In this chapter you will learn more about whole numbers, and you will strengthen your skills to do calculations and to solve problems.

1.1 Properties of whole numbers 3

1.2 Calculations with whole numbers 7

1.3 Multiples, factors and prime factors 18

1.4 Solving problems 24

A table of products

\times

23

46

79

88

117

124

178

276

348

8

184

368

632

704

936

992

1 424

2 208

2 784

18

414

828

1 422

1 584

2 106

2232

3 204

4 968

6 264

27

621

1 242

2 133

2 376

3 159

3348

4 806

7 452

9 396

34

782

1564

2 686

2 992

3 978

4216

6 052

9 384

11 832

47

1 081

2 162

3 713

4 136

5 499

5828

8 366

12 972

16 356

56

1 288

2 576

4 424

4 928

6 552

6944

9 968

15 456

19 488

67

1 541

3 082

5 293

5 896

7 839

8308

11 926

18 492

23 316

78

1 794

3 588

6 162

6 864

9 126

9672

13 884

21 528

27 144

84

1 932

3 864

6 636

7 392

9 828

10 416

14 952

23 184

29 232

93

2 139

4 278

7 347

8 184

10 881

11 532

16 554

25 668

32 364

Tables of sums

+

154

235

331

456

572

638

764

885

921

228

382

463

559

684

800

866

992

1 113

1 149

367

521

602

698

823

939

1 005

1 131

1 252

1 288

473

627

708

804

929

1 045

1 111

1 237

1 358

1 394

539

693

774

870

995

1 111

1 177

1 303

1 424

1 460

677

831

912

1 008

1 133

1 249

1 315

1 441

1 562

1 598

743

897

978

1 074

1 199

1 315

1 381

1 507

1 628

1 664

829

983

1 064

1 160

1 285

1 401

1 467

1 593

1 714

1 750

947

1 101

1 182

1 278

1 403

1 519

1 585

1 711

1 832

1 868

+

3154

7235

5331

9456

2572

3638

4764

2885

4921

228

3 382

7 463

5 559

9 684

2 800

3 866

4 992

3 113

5 149

367

3 521

7 602

5 698

9 823

2 939

4 005

5 131

3 252

5 288

473

3 627

7 708

5 804

9 929

3 045

4 111

5 237

3 358

5 394

539

3 693

7 774

5 870

9 995

3 111

4 177

5 303

3 424

5 460

677

3 831

7 912

6 008

10 133

3 249

4 315

5 441

3 562

5 598

743

3 897

7 978

6 074

10 199

3 315

4 381

5 507

3 628

5 664

829

3 983

8 064

6 160

10 285

3 401

4 467

5 593

3 714

5 750

947

4 101

8 182

6 278

10 403

3 519

4 585

5 711

3 832

5 868

1 Whole numbers

1.1 Properties of whole numbers

the commutative property of addition and multiplication

1. Which of the following calculations would you choose to calculate the number of yellow beads in this pattern? Do not do any calculations now, just make a choice.

93379.png

(a) 7 + 7 + 7 + 7 + 7

(b) 10 + 10 + 10 + 10 + 10 + 10 + 10

(c) 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5

(d) 5 + 5 + 5 + 5 + 5 + 5 + 5

(e) 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7

(f) 10 + 10 + 10 + 10 + 10

My choice:


2. (a) How many red beads are there in the pattern, and how many yellow beads?


(b) How many beads are there in the pattern in total?


3. (a) Which expression describes what you did to calculate the total number of beads:

70 + 50 or 50 + 70?


(b) Does it make a difference?


(c) Which expression describes what you did to calculate the number of red beads:

7 \times 10 or 10 \times 7?


(d) Does it make a difference?


We say: addition and multiplication are commutative. The numbers can be swopped around and their order does not change the answer. This does not work for subtraction and division, however.

4. Calculate each of the following:

5 \times 8


10 \times 8


12 \times 8


8 \times 12


6 \times 8


3 \times 7


6 \times 7


7 \times 6


the associative property of addition and multiplication

Lebogang and Nathi both have to calculate 25 \times 24.

Lebogang calculates 25 \times 4 and then multiplies by 6.

Nathi calculates 25 \times 6 and then multiplies by 4.

1. Will they get the same answer or not?


If three or more numbers have to be multiplied, it does not matter which two of the numbers are multiplied first.

2. Do the following calculations. Do not use a calculator now.

(a) 4 + 7 + 5 + 6


(b) 7 + 6 + 5 + 4


(c) 6 + 5 + 7 + 4


(d) 7 + 5 + 4 + 6


3. (a) Is addition associative?


(b) Illustrate your answer with an example.


4. Find the value of each expression by working in the easiest possible way.

(a) 2 \times 17 \times 5


(b) 4 \times 7 \times 5


(c) 75 + 37 + 25


(d) 60 + 87 + 40 + 13


5. What must you add to each of the following numbers to get 100?

82 44 56 78 24 89 77


6. What must you multiply each of these numbers by to get 1 000?

250 125 25 500 200 50


7. Calculate each of the following. Note that you can make the work very easy by being smart in deciding how to group the operations.

(a) 82 + 54 + 18 + 46 + 237 (b) 24 + 89 + 44 + 76 + 56 + 11

= (82 + 18) + (54 + 46) + 237 = 437

= (24 + 76) + (89 + 11) + (44 + 56) = 300

(c) 25 \times (86 \times 4) (d) 32 \times 125

= (25 \times 4) \times 86 = 100 \times 86 = 8 600

= (8 \times 125) \times 4 = 1 000 \times 4 = 4 000

more conventions and the distributive property

The distributive property is a useful property because it allows us to do this:

92648.png 

Both answers are 18. Notice that we have to use brackets in the first example to show that the addition operation must be done first. Otherwise, we would have done the multiplication first. For example, the expression 3 \times 2 + 4 means "multiply 3 by 2; then add 4". It does not mean "add 2 and 4; then multiply by 3".

The expression 4 + 3 \times 2 also means "multiply 3 by 2; then add 4".

If you wish to specify that addition or subtraction should be done first, that part of the expression should be enclosed in brackets.

The distributive property can be used to break up a difficult multiplication into smaller parts. For example, it can be used to make it easier to calculate 6 \times 204:

6 \times 204 can be rewritten as 6 \times (200 + 4) (Remember the brackets!)

= 6 \times 200 + 6 \times 4

= 1 200 + 24

= 1 224

Multiplication can also be distributed over subtraction, for example to calculate 7 \times 96:

7 \times 96 = 7 \times (100 - 4)

= 7 \times 100 - 7 \times 4

= 700 - 28

= 672

1. Here are some calculations with answers. Rewrite them with brackets to make all the answers correct.

(a) 8 + 6 \times 5 = 70 (b) 8 + 6 \times 5 = 38

(8 + 6) \times 5 = 70

8 + (6 \times 5) = 38

(c) 5 + 8 \times 6 - 2 = 52 (d) 5 + 8 \times 6 - 2 = 76

(5 + 8) \times (6 - 2) = 52

(5 + 8) \times 6 - 2 = 76

(e) 5 + 8 \times 6 - 2 = 51 (f) 5 + 8 \times 6 - 2 = 37

5 + (8 \times 6) - 2 = 51

5 + 8 \times (6 - 2) = 37

2. Calculate the following:

(a) 100 \times (10 + 7) (b) 100 \times 10 + 100 \times 7

= 1 000 + 700 = 1 700

= 1 000 + 700 = 1 700

(c) 100 \times (10 - 7) (d) 100 \times 10 - 100 \times 7

= 1 000 - 700 = 300

= 1 000 - 700 = 300

3. Complete the table.

\times

8

5

4

9

7

3

6

2

10

11

12

7

56

35

28

63

49

21

42

14

70

77

84

3

24

15

12

27

21

9

18

6

30

33

36

9

72

45

36

81

63

27

54

18

90

99

108

5

40

25

20

45

35

15

30

10

50

55

60

8

64

40

32

72

56

24

48

16

80

88

96

6

48

30

24

54

42

18

36

12

60

66

72

4

32

20

16

36

28

12

24

8

40

44

48

2

16

10

8

18

14

6

12

4

20

22

24

10

80

50

40

90

70

30

60

20

100

110

120

12

96

60

48

108

84

36

72

24

120

132

144

11

88

55

44

99

77

33

66

22

110

121

132

4. Use the various mathematical conventions for numerical expressions to make these calculations easier. Show all your working.

(a) 18 \times 50 (b) 125 \times 28 (c) 39 \times 220



















(d) 443 + 2 100 + 557 (e) 318 + 650 + 322 (f) 522 + 3 003 + 78







=1 000 + 2 100 = 3 100






1.2 Calculations with whole numbers

estimating, approximating and rounding

1. Try to give answers that you trust to these questions, without doing any calculations with the given numbers.

(a) Is 8 \times 117 more than 2 000 or less than 2 000?


than 2 000



(b) Is 27 \times 88 more than 3 000 or less than 3 000?


than 3 000



(c) Is 18 \times 117 more than 3 000 or less than 3 000?


than 3 000



(d) Is 47 \times 79 more than 3 000 or less than 3 000?


than 3 000



What you have done when you tried to give answers to questions 1(a) to (d), is called estimation. To estimate is to try to get close to an answer without

An estimate may also be called an approximation.

actually doing the required calculations with the given numbers.

2. Look at question 1 again.

(a) The numbers 1 000, 2 000, 3 000, 4 000, 5 000, 6 000, 7 000, 8 000, 9 000 and 10 000 are all multiples of a thousand. In each case, write down the multiple of 1 000 that you think is closest to the answer. Write it on the short dotted line. The numbers you write down are called estimates.

(b) In some cases you may think that you may achieve a better estimate by adding 500 to your estimate, or subtracting 500 from it. If so, you may add or subtract 500.

(c) If you wish, you may write what you believe is an even better estimate by adding or subtracting some hundreds.

3. (a) Use a calculator to find the exact answers for the calculations in question 1, or look up the answers in one of the tables on page 2.

The difference between an estimate and the actual answer is called the error.

Calculate the error in your last approximation of each of the answers in question 1.

(b) What was your smallest error?


4. Think again about what you did in question 2. In 2(a) you tried to approximate the answers to the nearest 1 000. In 2(c) you tried to approximate the answers to the nearest 100. Describe what you tried to achieve in question 2(b). Sample answer:


5. Estimate the answers for each of the following products and sums. Try to approximate the answers for the products to the nearest thousand, and for the sums to the nearest hundred. Use the first line in each question to do this.

(a) 84 \times 178


(b) 677 + 638


(c) 124 \times 93


(d) 885 + 473


(e) 79 \times 84


(f) 921 + 367


(g) 56 \times 348


(h) 764 + 829


6. Use a calculator to find the exact answers for the calculations in question 5, or look up the answers in the tables on page 2. Calculate the error in each of your approximations. Use the second line in each question to do this.

Calculating with "easy" numbers that are close to given numbers is a good way to obtain approximate answers, for example:

7. Calculate with "easy" numbers close to the given numbers to produce approximate answers for each product below. Do not use a calculator. When you have made your approximations, look up the precise answers in the top table on page 2.

(a) 78 \times 46


(b) 67 \times 88


80 \times 50 = 4 000

70 \times 90 = 6 300

(c) 34 \times 276


(d) 78 \times 178


30 \times 280 = 8 400

80 \times 180 = 14 400

rounding off and compensating

1. (a) Approximate the answer for 386 + 3 435, by rounding both numbers off to the nearest hundred, and adding the rounded numbers.

The word compensate means to do things that will remove damage.


(b) Because you rounded 386 up to 400, you introduced an error of 14 in your approximate answer. What error did you introduce by rounding 3 435 down to 3 400?


(c) What was the combined (total) error introduced by rounding both numbers off before calculating?


(d) Use your knowledge of the total error to correct your approximate answer, so that you have the correct answer for 386 + 3 435.


What you have done in question 1 to find the correct answer for 386 + 3 435 is called rounding off and compensating. By rounding the numbers off you introduced errors. You then compensated for the errors by making adjustments to your answer.

2. Round off and compensate to calculate each of the following accurately:

(a) 473 + 638 Sample answer: (b) 677 + 921 Sample answer:

470 + 640 = 1 110

680 + 920 = 1 600

1 110 + 3 - 2 = 1 111

1 600 - 3 + 1 = 1 598

Subtraction can also be done in this way. For example, to calculate R5 362 - R2 687, you may round R2 687 up to R3 000. The calculation can proceed as follows:

This means that R5 362 - R2 687 = R2 675, because

R5 362 - R2 687 = (R5 362 + R313) - (R2 687 + R313).

adding numbers in parts written in columns

Numbers can be added by thinking of their parts as we say the numbers.

For example, we say 4 994 as four thousand nine hundred and ninety-four. This can be written in expanded notation as 4 000 + 900 + 90 + 4.

Similarly, we can think of 31 837 as 30 000 + 1 000 + 800 + 30 + 7.

31 837 + 4 994 can be calculated by working with the various kindsof parts separately. To make this easy, the numbers can be written below each other so that the units are below the units, the tens below the tens and so on, as shown on the right.

31 837

4 994

We write only this:

In your mind you can see this:

31 837

30 000

1 000

800

30

7

4 994

4 000

900

90

4

The numbers in each column can be added to get a new set of numbers.

31 837

30 000

1 000

800

30

7

4 994

4 000

900

90

4

11

11

120

120

1 700

1 700

5 000

5 000

30 000

30 000

36 831

It is easy to add the new set of numbers to get the answer.

The work may start with the 10 000s or any other parts. Starting with the units as shown above makes it possible to do more of the work mentally, and write less, as shown below.

31 837

To achieve this, only the units digit 1 of the 11 is written in the first step. The 10 of the 11 is remembered and added to the 30 and 90 of the tens column, to get 130.

4 994

36 831

We say the 10 is carried from the units column to the tens column. The same is done when the tens parts are added to get 130: only the digit "3" is written (in the tens column, so it means 30), and the 100 is carried to the next step.

1. Calculate each of the following without using a calculator:

(a) 4 638 + 2 667 (b) 748 + 7 246

4 638

748

2667

7246

7 305

7 994

2. Impilo Enterprises plans a new computerised training facility in their existing building. The training manager has to keep the total expenditure budget under R1 million. This is what she has written so far:

Architects and builders

Painting and carpeting

Security doors and blinds

Data projector

25 new secretary chairs

24 desks for work stations

1 desk for presenter

25 new computers

12 colour laser printers

R102 700

R 42 600

R 52 000

R 4 800

R 50 400

R123 000

R 28 000

R300 000

R 38 980

Work out the total cost of all the items the training manager has budgeted for.








3. Calculate each of the following without using a calculator:

(a) 7 828 + 6 284 (b) 7 826 + 888 + 367

= 14 112

= 9 081

(c) 657 + 32 890 + 6 542 (d) 6 666 + 3 333 + 1

= 40 089

= 10 000

methods of subtraction

There are many ways to find the difference between two numbers. For example, to find the difference between 267 and 859 one may think of the numbers as they may be written on a number line.

90137.png 

We may think of the distance between 267 and 859 as three steps: from 267 to 300, from 300 to 800, and from 800 to 859. How big are each of these three steps?

90128.png 

The above shows that 859 - 267 is 33 + 500 + 59.

1. Calculate 33 + 500 + 59 to find the answer for 859 - 267.


2. Calculate each of the following. You may think of working out the distance between the two numbers as shown above, or use any other method you prefer. Do not use a calculator now.

(a) 823 - 456 (b) 1 714 - 829

44 + 300 + 23

71 + 800 + 14

= 367

= 885

(c) 3 045 - 2 572 (d) 5 131 - 367

28 + 400 + 45

33 + 600 + 4 000 + 131

= 473

= 4 764

You can use the tables of sums on page 2 to check your answers for question 2.

Like addition, subtraction can also be done by working with the different parts in which we say numbers. For example, 8 764 - 2 352 can be calculated as follows:

8 thousand - 2 thousand = 6 thousand

7 hundred - 3 hundred = 4 hundred

6 tens - 5 tens = 1 ten

4 units - 2 units = 2 units

So, 8 764 - 2 352 = 6 412

Subtraction by parts is more difficult in some cases, for example 6 213 - 2 758:

6 000 - 2 000 = 4 000. This step is easy, but the following steps cause problems:

200 - 700 = ?

10 - 50 = ?

3 - 8 = ?

Fortunately, the parts and sequence of work may be rearranged to overcome these problems, as shown below:

One way to overcome these problems is to work with negative numbers:200 - 700 = (-500)10 - 50 = (-40)3 - 8 = (-5)4 000 - 500 \rightarrow 3 500 - 45 =

instead of

we may do

3 - 8 =

?

13 - 8 =


"borrow" 10 from below

10 - 50 =

?

100 - 50 =


"borrow" 100 from below

200 - 700 =

?

1 100 - 700 =


"borrow" 1 000 from below

6 000 - 2 000 =

?

5 000 - 2 000 =


This reasoning can also be set out in columns:

instead of            

we may do        

but write only this

6 000

200

10

3

5 000

1 100

100

13

6

2

1

3

2 000

700

50

8

2 000

700

50

8

2

7

5

8

3 000

400

50

5

3

4

5

5

3. (a) Complete the above calculations and find the answer for 6 213 - 2 758.


(b) Use the borrowing technique to calculate 823 - 376 and 6 431 - 4 968.







4. Check your answers in question 3(b) by doing addition.


376 4 968


With some practice, you can learn to subtract using borrowing without writing all the steps. It is convenient to work in columns, as shown on the right for calculating 6 213 - 2 758.

In fact, by doing more work mentally, you may learn to save more paper by writing even less as shown below.

6 213

2758

5

50

400

3000

3 455

6 213

2758

3 455

Do not use a calculator when you do question 5, because the purpose of this work is for you to come to understand methods of subtraction. What you will learn here will later help you to understand algebra better.

5. Calculate each of the following:

(a) 7 342 - 3 877 (b) 8 653 - 1 856 (c) 5 671 - 4 528







You may use a calculator to do questions 6 and 7.

6. Estimate the difference between the two car prices in each case to the nearest R1 000 or closer. Then calculate the difference.

(a) R102 365 and R98 128 (b) R63 378 and R96 889

estimated: R4 000 or R4 200

estimated: R34 000 or R33 500

calculated: R4 237

calculated: R33 511

7. First estimate the answers to the nearest 100 000 or 10 000 or 1 000. Then calculate.

(a) 238 769 -141 453 (b) 856 333 - 739 878 (c) 65 244 - 39 427













a method of multiplication

7 \times 4 598 can be calculated in parts, as shown here:

4

5

9

8

7

5

6

6

3

0

3

5

0

0

2

8

0

0

0

3

2

1

8

6

7 \times 4 000 = 28 000

7 \times 500 = 3 500

7 \times 90 = 630

7 \times 8 = 56

The four partial products can now be added to get the answer, which is 32 186. It is convenient to write the work in vertical columns for units, tens, hundreds and so on, as shown on the right.

The answer can be produced with less writing, by "carrying" parts of the partial answers to the next column, when working from right to left in the columns.

4

5

9

8

7

3

2

1

8

6

Only the 6 of the product 7 \times 8 is written down instead of 56. The 50 is kept in mind, and added to the 630 obtained when 7 \times 90 is calculated in the next step.

1. Calculate each of the following. Do not use a calculator now.

(a) 27 \times 649 (b) 75 \times 1 756 (c) 348 \times 93







2. Use your calculator to check your answers for question 1. Redo the questions for which you had the wrong answers.

3. Calculate each of the following. Do not use a calculator now.

(a) 67 \times 276 (b) 84 \times 178

4. Use the product table on page 2 or a calculator to check your answers for question 3. Redo the questions for which you had the wrong answers.

long division

1. The municipal head gardener wants to buy young trees to plant along the main street of the town. The young trees cost R27 each, and an amount of R9 400 has been budgeted for trees. He needs 324 trees. Do you think he has enough money?



2. (a) How much will 300 trees cost?


(b) How much money will be left if 300 trees are bought?


(c) How much money will be left if 20 more trees are bought?


The municipal gardener wants to work out exactly how many trees, at R27 each, he can buy with the budgeted amount of R9 400. His thinking and writing are described below.

Step 1

What he writes:

What he thinks:

27

9400

Iwant to find out how many chunks of 27 there are in 9 400.

Step 2

What he writes:

What he thinks:

300

Ithink there are at least 300 chunks of 27 in 9 400.

27

9400

8100

300 \times 27 = 8 100. I need to know how much is left over.

1300

Iwant to find out how many chunks of 27 there are in 1 300.

Step 3 (He has to rub out the one "0" of the 300 on top, to make space.)

What he writes:

What he thinks:

340

Ithink there are at least 40 chunks of 27 in 1 300.

27

9400

8100

1300

1080

40 \times 27 = 1 080. I need to know how much is left over.

220

Iwant to find out how many chunks of 27 there are in 220.Perhaps I can buy some extra trees.

Step 4 (He rubs out another "0".)

What he writes:

What he thinks:

348

Ithink there are at least 8 chunks of 27 in 220.

27

9400

8100

1300

1080

220

216

8\times 27 = 216

4

So, I can buy 348 young trees and will have R4 left.

Do not use a calculator to do questions 3 and 4. The purpose of this work is for you to develop a good understanding of how division can be done. Check all your answers by doing multiplication.

3. (a) Graham bought 64 goats, all at the same price. He paid R5 440 in total. What was the price for each goat? Your first step can be to work out how much he would have paid if he paid R10 per goat, but you can start with a bigger step if you wish.







(b) Mary has R2 850 and she wants to buy candles for her sister's wedding reception. The candles cost R48 each. How many candles can she buy?









4. Calculate each of the following, without using a calculator:

(a) 7 234 \div 48 (b) 3 267 \div 24

150 with remainder 34

136 with remainder 3

(c) 9 500 \div 364 (d) 8 347 \div 24

26 with remainder 36

347 with remainder 19

1.3 Multiples, factors and prime factors

multiples and factors

1. The numbers 6; 12; 18; 24; ... are multiples of 6.

The numbers 7; 14; 21; 28; ... are multiples of 7.

If n is a natural number, 6n represents the multiples of 6.

(a) What is the 100th number in each sequence above?


(b) Is 198 a number in the first sequence?


(c) Is 175 a number in the second sequence?


Of which numbers is 20 a multiple?

20 = 1 \times 20 = 2 \times 10 = 4 \times 5 = 5\times 4 = 10 \times 2 = 20 \times 1

Factors come in pairs. The following pairs are factors of 20:

20 is a multiple of 1; 2; 4; 5; 10 and 20 and all of these numbers are factors of 20.

87332.png 

2. A rectangle has an area of 30 cm. What are the possible lengths of the sides of the rectangle in centimetres if the lengths of the sides are natural numbers?



3. Are 4; 8; 12 and 16 factors of 48? Simon says that all multiples of 4 smaller than 48 are factors of 48. Is he right?



4. We have defined factors in terms of the product of two numbers. What happens if we have a product of three or more numbers, for example 210 = 2 \times 3 \times 5 \times 7?

(a) Explain why 2; 3; 5 and 7 are factors of 210.


(b) Are 2 \times 3; 3 \times 5; 5 \times 7; 2 \times 5 and 2 \times 7 factors of 210?


(c) Are 2 \times 3 \times 5; 3 \times 5 \times 7 and 2 \times 5 \times 7 factors of 210?



5. Is 20 a factor of 60? What factors of 20 are also factors of 60?


prime numbers and composite numbers

1. Express each of the following numbers as a product of as many factors as possible, including repeated factors. Do not use 1 as a factor.

The number 36 can be formed as 2 \times 2 \times 3 \times 3. Because 2 and 3 are used twice, they are called repeated factors of 36.

(a) 66 (b) 67

2 \times 33; 3 \times 22; 6 \times 11; 2 \times 3 \times 11

No other factor

(c) 68 (d) 69

2 \times 34; 4 \times 17; 2 \times 2 \times 17

3 \times 23

(e) 70 (f) 71

2 \times 35; 5 \times 14; 7 \times 10; 2 \times 5 \times 7

No other factors

(g) 72 (h) 73

2 \times 36; 3 \times 24; 4 \times 18; 6 \times 12;

No other factors

8 \times 9; 2 \times 2 \times 2 \times 3 \times 3

2. Which of the numbers in question 1 cannot be expressed as a product of two whole numbers, except as the product 1\times the number itself?


Anumber that cannot be expressed as a product of two whole numbers, except as the product1\times the number itself, is called a prime number.

3. Which of the numbers in question 1 are prime?


Composite numbers are natural numbers with more than two different factors. The sequence of composite numbers is 4; 6; 8; 9; 10; 12; ...

4. Are the statements below true or false? If you answer "false", explain why.

(a) All prime numbers are odd numbers.


(b) All composite numbers are even numbers.


(c) 1 is a prime number.


(d) If a natural number is not prime, then it is composite.


(e) 2 is a composite number.


(f) 785 is a prime number.


(g) A prime number can only end in 1; 3; 7 or 9.


(h) Every composite number is divisible by at least one prime number.


5. We can find out whether a given number is prime by systematically checking whether the primes 2; 3; 5; 7; 11; 13; … are factors of the given number or not.

To find possible factors of 131, we need to consider only the primes 2; 3; 5; 7 and 11. Why not 13; 17; 19; …?


6. Determine whether the following numbers are prime or composite. If the number is composite, write down at least two factors of the number (besides 1 and the number itself).

(a) 221 (b) 713

Composite: 13; 17

Composite: 23; 31

prime factorisation

To find all the factors of a number you can write the number as the product of prime factors, first by writing it as the product of two convenient (composite) factors and then by splitting these factors into smaller factors until all factors are prime. Then you take all the possible combinations of the products of the prime factors.

Every composite number can be expressed as the product of prime factors and this can happen in only one way.

Example: Find the factors of 84.

Write 84 as the product of prime factors by starting with different known factors:

84 = 4 \times 21 or 84 = 7 \times 12 or 84 = 2 \times 42

= 2 \times 2 \times 3 \times 7 = 7 \times 3 \times 4 = 2 \times 6 \times 7

= 7 \times 3 \times 2 \times 2 = 2 \times 2 \times 3 \times 7

A more systematic way of finding the prime factors of a number would be to start with the prime numbers and try the consecutive prime numbers 2; 3; 5; 7; ... as possible factors. The work may be set out as shown below.

2

1 430

3

2 457

5

715

3

819

11

143

3

273

13

13

7

91

1

13

13

1

1 430 = 2 \times 5 \times 11 \times 13 2 457 = 3 \times 3 \times 3 \times 7 \times 13

We can use exponents to write the products of prime factors more compactly as products of powers of prime factors.

2 457 = 3 \times 3 \times 3 \times 7 \times 13 = 33 \times 7 \times 13

72 = 2 \times 2 \times 2 \times 3 \times 3 = 23 \times 32

1 500 = 2 \times 2 \times 3 \times 5 \times 5 \times 5 = 22 \times 3 \times 53

1. Express the following numbers as the product of powers of primes:

(a) 792 =


(b) 444 =


2. Find the prime factors of the numbers below.

2

28

32

124

36

42

345

182

14



common multiples and factors

1. Is 4 \times 5 a multiple of 4?


Is 4 \times 5 a multiple of 5?


2. Comment on the following statement: The product of numbers is a multiple of each of the numbers in the product.


We use common multiples when fractions with different denominators are added.

To add 86725.png + 86716.png, the common denominator is 3 \times 4, so the sum becomes 86709.png + 86700.png.

In the same way, we could use 6 \times 8 = 48 as a common denominator to add 86692.png + 86685.png, but

24 is the lowest common multiple (LCM) of 6 and 8.

Prime factorisation makes it easy to find the lowest common multiple or highest common factor.

When we simplify a fraction, we divide the same number into the numerator and the denominator. For the simplest fraction, use the highest common factor (HCF) to divide into both numerator and denominator.

The HCF is divided into the numerator and the denominator to write the fraction in its simplest form.

So 86650.png = 86639.png = 86632.png 

Use prime factorisation to determine the LCM and HCF of 32, 48 and 84 in a systematic way: 32 = 2 \times 2 \times 2 \times 2 \times 2 = 25

48 = 2 \times 2 \times 2 \times 2 \times 3 = 24 \times 3

84 = 2 \times 2 \times 3 \times 7 = 22 \times 3 \times 7

The LCM is a multiple, so all of the factors of all the numbers must divide into it.

All of the factors that are present in the three numbers must also be factors of the LCM, even if it is a factor of only one of the numbers. But because it has to be the lowest common multiple, no unnecessary factors are in the LCM.

The highest power of each factor is in the LCM, because then all of the other factors can divide into it. In 32, 48 and 84, the highest power of 2 is 25, the highest power of 3 is 3 and the highest power of 7 is 7.

LCM = 25 \times 3 \times 7 = 672

The HCF is a common factor. Therefore, for a factor to be in the HCF, it must be a factor of all of the numbers. 2 is the only number that appears as a factor of all three numbers. The lowest power of 2 is 22, so the HCF is 22.

3. Determine the LCM and the HCF of the numbers in each case.

(a) 24; 28; 42 (b) 17; 21; 35

24 = 23 \times 3

17 is prime

28 = 22 \times 7

21 = 3 \times 7

42 = 2 \times 3 \times 7

35 = 5 \times 7

HCF = 2

No HCF

LCM = 23 \times 3 \times 7 = 168

LCM = 3 \times 5 \times 17 = 255

(c) 75; 120; 200 (d) 18; 30; 45

75 = 3 \times 5 \times 5

18 = 2 \times 32

120 = 23 \times 3 \times 5

30 = 2 \times 3 \times 5

200 = 23 \times 52

45 = 32 \times 5

HCF = 5

HCF = 3

LCM = 23 \times 3 \times 52 = 600

LCM = 2 \times 32 \times 5 = 90

investigate prime numbers

  • •

You may use a calculator for this investigation.

1. Find all the prime numbers between 110 and 130.

2. Find all the prime numbers between 210 and 230.

3. Find the biggest prime number smaller than 1 000.

1.4 Solving problems

rate and ratio

You may use a calculator for the work in this section.

1. Tree plantations in the Western Cape are to be cut down in favour of natural vegetation. There are roughly 3 000 000 trees on plantations in the area and it is possible to cut them down at a rate of 15 000 trees per day with the labour available. How many working days will it take before all the trees will be cut down?




Instead of saying "… per day", people often say "at a rate of … per day". Speed is a way to describe the rate of movement.

The word per is often used to describe a rate and can mean for every, for, in each, in, out of, or every.

2. A car travels a distance of 180 km in 2 hours on a straight road. How many kilometres can it travel in 3 hours at the same speed?



3. Thobeka wants to order a book that costs $56,67. The rand-dollar exchange rate is R7,90 to a dollar. What is the price of the book in rands?





4. In pattern A below, there are 5 red beads for every 4 yellow beads.

86093.png 

Describe patterns B and C in the same way.




5. Complete the table to show how many screws are produced by two machines in different periods of time.

Number of hours

1

2

3

5

8

Number of screws at machine A

1 800

3 600

5 400

9 000

14 400

Number of screws at machine B

2 700

5 400

8 100

13 500

21 600

(a) How much faster is machine B than machine A?


(b) How many screws will machine B produce in the same time that it takes machine A to make 100 screws?


The patterns in question 4 can be described like this: In pattern A, the ratio of yellow beads to red beads is 4 to 5. This is written as 4 : 5. In pattern B, the ratio between yellow beads and red beads is 3 : 6. In pattern C the ratio is 2 : 7.

In question 5, machine A produces 2 screws for every 3 screws that machine B produces. This can be described by saying that the ratio between the production speeds of machines A and B is 2 : 3.

6. Nathi, Paul and Tim worked in Mr Setati's garden. Nathi worked for 5 hours, Paul for 4 hours and Tim for 3 hours. Mr Setati gave the boys R600 for their work. How should they divide the R600 among the three of them?



A ratio is a comparison of two (or more) quantities.

We use ratios to show how many times more, or less, one quantity is than another.

The number of hours that Nathi, Paul and Tim worked are in the ratio 5 : 4 : 3. To be fair, the money should also be shared in that ratio. That means that Nathi should receive 5 parts, Paul 4 parts and Tim 3 parts of the money. There were 12 parts, which means Nathi should

receive 86026.png of the total amount, Paul should get 86018.png and Tim should get 86010.png.

7. Ntabi uses 3 packets of jelly to make a pudding for 8 people. How many packets of jelly does she need to make a pudding for 16 people? And for 12 people?


8. Which rectangle is more like a square: a 3 \times 5 rectangle or a 6 \times 8 rectangle? Explain.

A 6 \times 8 rectangle, because is closer to 1 than is.

To increase 40 in the ratio 2 : 3 means that the 40 represents two parts and must be increased so that the new number represents 3 parts. If 40 represents two parts, 20 represents 1 part. The increased number will therefore be 20 \times 3 = 60.

Remember that if you multiply by 1, the number does not change. If you multiply by a number greater than 1, the number increases. If you multiply by a number smaller than 1, the number decreases.

9. (a) Increase 56 in the ratio 2 : 3.


(b) Decrease 72 in the ratio 4 : 3.


10. (a) Divide 840 in the ratio 3 : 4.


(b) Divide 360 in the ratio 1 : 2 : 3.



11. Some data about the performance of different athletes during a walking event is given below. Investigate the data to find out who walks fastest and who walks slowest. Arrange the athletes from the fastest walker to the slowest walker.

(a) First make estimates to do the investigation.

(b) Then use your calculator to do the investigation.

Athlete

A

B

C

D

E

F

Distance walked in m

2 480

4 283

3 729

6 209

3 112

5 638

Time taken in minutes

17

43

28

53

24

45









profit, loss, discount and interest

1. (a) How much is 1 eighth of R800?


(b) How much is 1 hundredth of R800?


(b) How much is 7 hundredths of R800?


Rashid is a furniture dealer. He buys a couch for R2 420. He displays the couch in his showroom with the price marked as R3 200. Rashid offers a discount of R320 to customers who pay cash.

The amount for which a dealer buys an article from a producer or manufacturer is called the cost price. The price marked on the article is called the marked price and the price of the article after discount is the selling price.

2. (a) What is the cost price of the couch in Rashid's furniture shop?


(b) What is the marked price?


(c) What is the selling price for a customer who pays cash?


(d) How much is 10 hundredths of R3 200?


The discount on an article is always less than the marked price of the article. In fact, it is only a fraction of the marked price. The discount of R320 that Rashid offers on the couch is 10 hundredths of the marked price.

Another word for hundredths is percentage, and the symbol for percentage is %. So we can say that Rashid offers a discount of 10%.

A percentage is a number of hundredths.

18% is 18 hundredths, and 25% is 25 hundredths.

%is a symbol for hundredths. 8% means 8 hundredths and 15% means 15 hundredths.

The symbol % is just a

variation of the 93575.png that is

used in the common fraction notation for hundredths.

8% is 93568.png.

A discount of 6% on an article can be calculated in two steps:

Step 1: Calculate 1 hundredth of the marked price (divide by 100).

Step 2: Calculate 6 hundredths of the marked price (multiply by 6).

3. Calculate a discount of 6% on each of the following marked prices of articles:

(a) R3 600 (b) R9 360

R3 600 \div 100 = R36

R9 360 \div 100 = R93,60

R36 \times 6 = R216

R93,60 \times 6 = R561,60

4. (a) How much is 1 hundredth of R700?


(b) A customer pays cash for a coat marked at R700. He is given R63 discount. How many hundredths of R700 is this?


(c) What is the percentage discount?


5. A client buys a blouse marked at R300 and she is given R36 discount for paying cash. Work as in question 4 to determine what percentage discount she was given.




You may use a calculator to do questions 6, 7 and 8.

6. A dealer buys an article for R7 500 and makes the price 30% higher. The article is sold at a 20% discount.

(a) What is the selling price of the article?

of 7 500 = R2 250 Marked price: R7 500 + R2 250 = R9 750

Selling price: of R9 750 = R7 800

(b) What is the dealer's percentage profit?

R7 800 - R7 500 = R300 \times = 4%

When a person borrows money from a bank or some other institution, he or she normally has to pay for the use of the money. This is called interest.

7. Sam borrows R7 000 from a bank at 14% interest for one year. How much does he have to pay back to the bank at the end of the period?



8. Jabu invests R5 600 for one year at 8% interest.

(a) What will the value of his investment be at the end of that year?


(b) At the end of the year Jabu does not withdraw the investment or the interest earned, but reinvests it for another year. How much will it be worth at the end of the second year?



(c) What will the value of Jabu's investment be after five years?


In this chapter you will work with whole numbers smaller than 0. These numbers are called negative numbers. The whole numbers larger than 0, 0 itself and the negative whole numbers together are called the integers. Mathematicians have agreed that negative numbers should have certain properties that would make them useful for various purposes. You will learn about these properties and how they make it possible to do calculations with negative numbers.

2.1 What is beyond 0? 31

2.2 Adding and subtracting with integers 35

2.3 Multiplying and dividing with integers 40

2.4 Squares, cubes and roots with integers 47

Do what you can.

5 - 0 = ?

5 - 7 = ?

5 + 5 = ?

5 - 1 = ?

5 - 6 = ?

5 + 4 = ?

5 - 2 = ?

5 - 5 = ?

5 + 3 = ?

5 - 3 = ?

5 - 4 = ?

5 + 2 = ?

5 - 4 = ?

5 - 3 = ?

5 + 1 = ?

5 - 5 = ?

5 - 2 = ?

5 + 0 = ?

5 - 6 = ?

5 - 1 = ?

5 + ? = ?

5 - 7 = ?

5 - 0 = ?

5 + ? = ?

5 - 8 = ?

5 - ? = ?

5 + ? = ?

5 - 9 = ?

5 -? = ?

5 + ? = ?

5 - 10 = ?

5 - ? = ?

5 + ? = ?

Choose a good plan to complete this table.

Plan A: Look at where the number 4 appears in the table. All the 4s lie on a diagonal, going down from left to right. Complete the other diagonals in the same way.

Plan B: Look at the number 13 in the table. It is on the right, in the fourth row from the top. It can be obtained by adding the two numbers indicated by arrows. Complete all the cells by adding numbers from the yellow column and blue row in this way.

Plan C: In each row, add 1 to go right and subtract 1 to go left.

Plan D: In each column, add 1 to go up and subtract 1 to go down.

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

4

7

4

6

–3

–2

–1

0

4

5

13

4

–5

–4

3

4

2

4

1

4

–8

–7

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

7

8

–1

4

–2

4

–3

4

–4

4

–5

–6

–7

0

–8

0

84680.png
84681.png

2 Integers

2.1 What is beyond 0?

why people decided to have negative numbers

On the right, you can see how Jimmy prefers to work when doing calculations such as 542 + 253.

500 + 200 = 700

40 + 50 = 90

2 + 3 = 5

700 + 90 + 5 = 795

He tries to calculate 542 - 253 in a similar way:

500 - 200 = 300

40 - 50 = ?

Jimmy clearly has a problem. He reasons as follows:

Ican subtract 40 from 40; that gives 0. But then there is still 10 that I have to subtract.

He decides to deal with the 10 that he still has to subtract later, and continues:

500 - 200 = 300

40 - 50 = 0, but there is still 10 that I have to subtract.

2 - 3 = 0, but there is still 1 that I have to subtract.

1. (a) What must Jimmy still subtract, and what will his final answer be?



(b) When Jimmy did another subtraction problem, he ended up with this writing at one stage:

600 and (-)50 and (-)7

What do you think is Jimmy's final answer for this subtraction problem?


About 500 years ago, some mathematicians proposed that a "negative number" may be used to describe the result in a situation such as in Jimmy's subtraction problem above, where a number is subtracted from a number smaller than itself.

For example, we may say 10 - 20 = (-10)

This proposal was soon accepted by other mathematicians, and it is now used all over the world.

Mathematicians are people who do mathematics for a living. Mathematics is their profession, like health care is the profession of nurses and medical doctors.

2. Calculate each of the following:

(a) 16 - 20


(b) 16 - 30


(c) 16 - 40


(d) 16 - 60


(e) 16 - 200


(f) 5 - 1 000


3. Some numbers are shown on the lines below. Fill in the missing numbers.

84456.png 

84447.png 

The following statement is true if the number is 5:

The numbers 1; 2; 3; 4 etc. are called the natural numbers. The natural numbers, 0 and the negative whole numbers together are called the integers.

15 - (a certain number) = 10

A few centuries ago, some mathematicians decided they wanted to have numbers that will also make sentences like the following true:

15 + (a certain number) = 10

But to go from 15 to 10 you have to subtract 5.

The number we need to make the sentence 15 + (a certain number) = 10 true must have the following strange property:

If you add this number, it should have the same effect as to subtract 5.

Now the mathematicians of a few centuries ago really wanted to have numbers for which such strange sentences would be true. So they thought:

Let us decide, and agree amongst ourselves, that the number we call negative 5 will have the property that if you add it to another number, the effect will be the same as when you subtract the natural number 5.

This means that the mathematicians agreed that 15 + (-5) is equal to 15 - 5.

Stated differently, instead of adding negative 5 to a number, you may subtract 5.

Adding a negative number has the same effect as subtracting a natural number.

For example: 20 + (-15) = 20 - 15 = 5

4. Calculate each of the following:

(a) 500 + (-300)


(b) 100 + (-20) + (-40)


(c) 500 + (-200) + (-100)


(d) 100 + (-60)


5. Make a suggestion of what the answer for (-20) + (-40) should be. Give reasons for your suggestion.


6. Continue the lists of numbers below to complete the table.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

10

100

3

-3

-20

150

0

9

90

6

-6

-18

125

-5

8

80

9

-9

-16

100

-10

7

70

12

-12

-14

75

-15

6

60

15

-15

-12

50

-20

5

50

18

-18

-10

25

-25

4

40

21

-21

-8

0

-30

3

30

24

-24

-6

-25

-35

2

20

27

-27

-4

-50

-40

1

10

30

-30

-2

-75

-45

-1

-10

33

-33

2

-100

-50

-2

-20

36

-36

4

-125

-55

-3

-30

39

-39

6

-150

-60

-4

-40

42

-42

8

-175

-65

The following statement is true if the number is 5:

15 +(a certain number) = 20

What properties should a number have so that it makes the following statement true?

15 - (a certain number) = 20

To go from 15 to 20 you have to add 5. The number we need to make the sentence 15 - (a certain number) = 20 true must have the following property:

If you subtract this number, it should have the same effect as to add 5.

Let us agree that 15 - (-5) is equal to 15 + 5.

Stated differently, instead of subtracting negative 5 from a number, you may add 5.

Subtracting a negative number has the same effect as adding a natural number.

For example: 20 - (-15) = 20 + 15 = 35

7. Calculate.

(a) 30 - (-10)


(b) 30 + 10


(c) 30 + (-10)


(d) 30 - 10


(e) 30 - (-30)


(f) 30 + 30


(g) 30 + (-30)


(h) 30 - 30


You probably agree that

5 + (-5) = 0 10 + (-10) = 0 and 20 + (-20) = 0

We may say that for each "positive" number there is a corresponding or opposite negative number. Two positive and negative numbers that correspond, for example 3 and (-3), are called additive inverses. They wipe each other out when you add them.

What may each of the following be equal to?

(-8) + 5

(-5) + (-8)

When you add any number to its additive inverse, the answer is 0 (the additive property of 0). For example, 120 + (-120) = 0.

8. Write the additive inverse of each of the following numbers:

(a) 24


(b) -24


(c) -103


(d) 2 348


The idea of additive inverses may be used to explain why 8 + (-5) is equal to 3:

8 + (-5) = 3 + 5 + (-5) = 3 + 0 = 3

83958.jpg

9. Use the idea of additive inverses to explain why each of these statements is true:

(a) 43 + (-30) = 13 (b) 150 + (-80) = 70

13 + 30 - 30 = 13

70 + 80 + (-80) = 70

STATEMENTS THAT ARE TRUE FOR MANY DIFFERENT NUMBERS

  • •

For how many different pairs of numbers can the following statement be true, if only natural (positive) numbers are allowed?

a number + another number = 10

For how many different pairs of numbers can the statement be true if negative numbers are also allowed?

2.2 Adding and subtracting with integers

adding can make less and subtraction can make more

1. Calculate each of the following:

(a) 10 + 4 + (-4)


(b) 10 + (-4) + 4


(c) 3 + 8 + (-8)


(d) 3 + (-8) + 8


Natural numbers can be arranged in any order to add and subtract them. This is also the case for integers.

The numbers 1; 2; 3; 4; etc. that we use to count, are called natural numbers.

2. Calculate each of the following:

(a) 18 + 12


(b) 12 + 18


(c) 2 + 4 + 6


(d) 6 + 4 + 2


(e) 2 + 6 + 4


(f) 4 + 2 + 6


(g) 4 + 6 + 2


(h) 6 + 2 + 4


(i) 6 + (-2) + 4


(j) 4 + 6 + (-2)


(k) 4 + (-2) + 6


(l) (-2) + 4 + 6


(m) 6 + 4 + (-2)


(n) (-2) + 6 + 4


(o) (-6) + 4 + 2


3. Calculate each of the following:

(a) (-5) + 10


(b) 10 + (-5)


(c) (-8) + 20


(d) 20 - 8


(e) 30 + (-10)


(f) 30 + (-20)


(g) 30 + (-30)


(h) 10 + (-5) + (-3)


(i) (-5) + 7 + (-3) + 5


(j) (-5) + 2 + (-7) + 4


4. In each case, find the number that makes the statement true. Give your answer by writing a closed number sentence.

Statements like these are also called number sentences.

An incomplete number sentence, where some numbers are not known at first, is sometimes called an open number sentence:

8 - (a number) = 10

A closed number sentence is where all the numbers are known:

8 + 2 = 10

(a) 20 + (an unknown number) = 50


(b) 50 + (an unknown number) = 20


(c) 20 + (an unknown number) = 10


(d) (an unknown number) + (-25) = 50


(e) (an unknown number) + (-25) = -50



5. Use the idea of additive inverses to explain why each of the following statements is true:

(a) 43 + (-50) = -7


(b) 60 + (-85) = -25


6. Complete the table as far as you can.

(a)

(b)

(c)

5 - 8 = -3

5 + 8 = -3

8 - 3 = -3

5 - 7 = -3

5 + 7 = -3

7 - 3 = -3

5 - 6 = -3

5 + 6 = -3

6 - 3 = -3

5 - 5 = -3

5 + 5 = -3

5 - 3 = -3

5 - 4 = -3

5 + 4 = -3

4 - 3 = -3

5 - 3 = -3

5 + 3 = -3

3 - 3 = -3

5 - 2 = -3

5 + 2 = -3

2 - 3 = -3

5 - 1 = -3

5 + 1 = -3

1 - 3 = -3

5 - 0 = -3

5 + 0 = -3

0 - 3 = -3

5 - (-1) = -3

5 + (-1) = -3

(-1) - 3 = -3

5 - (-2) = -3

5 + (-2) = -3

(-2) - 3 = -3

5 - (-3) = -3

5 + (-3) = -3

(-3) - 3 = -3

5 - (-4) = -3

5 + (-4) = -3

(-4) - 3 = -3

5 - (-5) = -3

5 + (-5) = -3

(-5) - 3 = -3

5 - (-6) = -3

5 + (-6) = -3

(-6) - 3 = -3

7. Calculate.

(a) 80 + (-60) (b) 500 + (-200) + (-200)

= 20

= 100

8. (a) Is 100 + (-20) + (-20) = 60, or does it equal something else?


(b) What do you think (-20) + (-20) is equal to?


9. Calculate.

(a) 20 - 20


(b) 50 – 20


(c) (-20) - (-20)


(d) (-50) - (-20)


10. Calculate.

(a) 20 - (-10)


(b) 100 - (-100)


(c) 20 + (-10)


(d) 100 + (-100)


(e) (-20) - (-10)


(f) (-100) - (-100)


(g) (-20) + (-10)


(h) (-100) + (-100)


11. Complete the table as far as you can.

(a)

(b)

(c)

5 - (-8) =-3

(-5) + 8 = -3

8 - (-3) = -3

5 - (-7) = -3

(-5) + 7 = -3

7 - (-3) = -3

5 - (-6) = -3

(-5) + 6 = -3

6 - (-3) = -3

5 - (-5) = -3

(-5) + 5 = -3

5 - (-3) = -3

5 - (-4) = -3

(-5) + 4 = -3

4 - (-3) = -3

5 - (-3) = -3

(-5) + 3 = -3

3 - (-3) = -3

5 - (-2) = -3

(-5) + 2 = -3

2 - (-3) = -3

5 - (-1) = -3

(-5) + 1 = -3

1 - (-3) = -3

5 - 0 = -3

(-5) + 0 = -3

0 - (-3) = -3

5 - 1 = -3

(-5) + (-1) = -3

(-1) - (-3) = -3

5 - 2 = -3

(-5) + (-2) = -3

(-2) - (-3) = -3

5 - 3 = -3

(-5) + (-3) = -3

(-3) - (-3) = -3

5 - 4 = -3

(-5) + (-4) = -3

(-4) - (-3) = -3

5 - 5 = -3

(-5) + (-5) = --3

(-5) - (-3) = -3

12. In each case, state whether the statement is true or false and give a numerical example to demonstrate your answer.

(a) Subtracting a positive number from a negative number has the same effect as adding the additive inverse of the positive number.


(b) Adding a negative number to a positive number has the same effect as adding the additive inverse of the negative number.


(c) Subtracting a negative number from a positive number has the same effect as subtracting the additive inverse of the negative number.


(d) Adding a negative number to a positive number has the same effect as subtracting the additive inverse of the negative number.


(e) Adding a positive number to a negative number has the same effect as adding the additive inverse of the positive number.


(f) Adding a positive number to a negative number has the same effect as subtracting the additive inverse of the positive number.


(g) Subtracting a positive number from a negative number has the same effect as subtracting the additive inverse of the positive number.


(h) Subtracting a negative number from a positive number has the same effect as adding the additive inverse of the negative number.


comparing integers and solving problems

1. Fill <, > or = into the block to make the relationship between the numbers true:

(a) -103

-99

(b) -699

-701

(c) 30

-30

(d) 10 - 7

-(10 - 7)

(e) -121

-200

(f) 12 - 5

-(12 + 5)

(g) -199

-110

2. At 5 a.m. in Bloemfontein the temperature was -5 °C. At 1 p.m., it was 19 °C. By how many degrees did the temperature rise?


3. A diver swims 150 m below the surface of the sea. She moves 75 m towards the surface. How far below the surface is she now?

-150 m + 75 m = -75 m


4. One trench in the ocean is 800 m deep and another is 2 200 m deep. What is the difference in their depths?



5. An island has a mountain which is 1 200 m high. The surrounding ocean has a depth of 860 m. What is the difference in height?


6. On a winter's day in Upington the temperature rose by 19 °C. If the minimum temperature was -4 °C, what was the maximum temperature?

-4 °C + 19 °C = 15 °C

2.3 Multiplying and dividing with integers

multiplication with integers

1. Calculate.

(a) -5 + -5 + -5 + -5 + -5 + -5 + -5 + -5 + -5 + -5


(b) -10 + -10 + -10 + -10 + -10


(c) -6 + -6 + -6 + -6 + -6 + -6 + -6 + -6


(d) -8 + -8 + -8 + -8 + -8 + -8


(e) -20 + -20 + -20 + -20 + -20 + -20 + -20


2. In each case, show whether you agree (✓) or disagree (✗) with the given statement.

(a) 10 \times (-5) = 50


(b) 8 \times (-6) = (-8) \times 6


(c) (-5) \times 10 = 5 \times (-10)


(d) 6 \times (-8) = -48


(e) (-5) \times 10 = 10 \times (-5)


(f) 8 \times (-6) = 48


(g) 4 \times 12 = -48


(h) (-4) \times 12 = -48


Multiplication of integers is commutative:

(-20) \times 5 = 5 \times (-20)

3. Is addition of integers commutative? Demonstrate your answer with three different examples.



4. Calculate.

(a) 20 \times (-10)


(b) (-5) \times 4


(c) (-20) \times 10


(d) 4 \times (-25)


(e) 29 \times (-20)


(f) (-29) \times (-2)


5. Calculate.

(a) 10 \times 50 + 10 \times (-30) (b) 50 + (-30)

= 200

= 20

(c) 10 \times (50 + (-30)) (d) (-50) + (-30)

= 200

= –80

(e) 10 \times (-50) + 10 \times (-30) (f) 10 \times ((-50) + (-30))

= –800

= –800

The product of two positive numbers is a positive number, for example 5 \times 6 = 30.

6. (a) Four numerical expressions are given below. Underline the expressions that you would expect to have the same answers. Do not do the calculations.

14 \times (23 + 58) 23 \times (14 + 58) 14 \times 23 + 14 \times 58 14 \times 23 + 58

(b) What property of operations is demonstrated by the fact that two of the above expressions have the same value?


7. Consider your answers for question 6.

(a) Does multiplication distribute over addition in the case of integers?


(b) Illustrate your answer with two examples.




8. Three numerical expressions are given below. Underline the expressions that you would expect to have the same answers. Do not do the calculations.

10 \times ((-50) - (-30)) 10 \times (-50) - (-30) 10 \times (-50) - 10 \times (-30)

9. Do the three sets of calculations given in question 8.



Your work in questions 5, 8 and 9 demonstrates that multiplication with a positive number distributes over addition and subtraction of integers. For example:

10 \times (5 + (-3)) = 10 \times 2 = 20 and 10 \times 5 + 10 \times (-3) = 50 + (-30) = 20

10 \times (5 - (-3)) = 10 \times 8 = 80 and 10 \times 5 - 10 \times (-3) = 50 - (-30) = 80

10. Calculate: (-10) \times (5 + (-3))


Now consider the question of whether multiplication with a negative number distributes over addition and subtraction of integers. For example, would (-10) \times 5 + (-10) \times (-3) also have the answer -20, as does (-10) \times (5 + (-3))?

11. What must (-10) \times (-3) be equal to, if we want (-10) \times 5 + (-10) \times (-3) to be equal to -20?



In order to ensure that multiplication distributes over addition and subtraction in the system of integers, we have to agree that

(a negative number) \times (a negative number) is a positive number,

for example (-10) \times (-3) = 30.

12. Calculate.

(a) (-10) \times (-5)


(b) (-10) \times 5


(c) 10 \times 5


(d) 10 \times (-5)


(e) (-20) \times (-10) + (-20) \times (-6) (f) (-20) \times ((-10) + (-6))

= 200 + 120 = 320

= -20 \times (-16) = 320

(g) (-20) \times (-10) - (-20) \times (-6) (h) (-20) \times ((-10) - (-6))

= 200 - 120 = 80

= -20 \times -4 = 80

Here is a summary of the properties of integers that make it possible to do calculations with integers:

division with integers

1. (a) Calculate 25 \times 8.


(b) How much is 200 \div 25?


(c) How much is 200 \div 8?


Division is the inverse of multiplication. Hence, if two numbers and the value of their product are known, the answers to two division problems are also known.

2. Calculate.

(a) 25 \times (-8) (b) (-125) \times 8

= -200

= -1 000

3. Use the work you have done for question 2 to write the answers for the following division questions:

(a) (-1 000) \div (-125)


(b) (-1 000) \div 8


(c) (-200) \div 25


(d) (-200) \div 8


4. Can you also work out the answers for the following division questions by using the work you have done for question 2?

(a) 1 000 \div (-125)


(b) (-1 000) \div (-8)


(c) (-100) \div (-25)


(d) 100 \div (-25)


When two numbers are multiplied, for example 30 \times 4 = 120, the word "product" can be used in various ways to describe the situation:

An expression that specifies division only, such as 30 \div 5, is called a quotient or a quotient expression. The answer obtained is also called the quotient of the two numbers. For example, 6 is called the quotient of 30 and 5.

5. In each case, state whether you agree or disagree with the statement, and give an example to illustrate your answer.

(a) The quotient of a positive and a negative integer is negative.


(b) The quotient of a positive and a positive integer is negative.


(c) The quotient of a negative and a negative integer is negative.


(d) The quotient of a negative and a negative integer is positive.


6. Do the necessary calculations to enable you to provide the values of the quotients.

(a) (-500) \div (-20) (b) (-144) \div 6

= 25

= -24

(c) 1 440 \div (-60) (d) (-1 440) \div (-6)

= -24

= 240

(e) -14 400 \div 600 (f) 500 \div (-20)

= -24

= -25

the associative properties of operations with integers

Multiplication of whole numbers is associative. This means that in a product with several factors, the factors can be placed in any sequence, and the calculations can be performed in any sequence. For example, the following sequences of calculations will all produce the same answer:

A. 2 \times 3, the answer of 2 \times 3 multiplied by 5, the new answer multiplied by 10

B. 2 \times 5, the answer of 2 \times 5 multiplied by 10, the new answer multiplied by 3

C. 10 \times 5, the answer of 10 \times 5 multiplied by 3, the new answer multiplied by 2

D. 3 \times 5, the answer of 3 \times 5 multiplied by 2, the new answer multiplied by 10

1. Do the four sets of calculations given in A to D to check whether they really produce the same answers.




D. 15 \times 2 = 30; 30 \times 10 = 30000

2. (a) If the numbers 3 and 10 in the calculation sequences A, B, C and D are replaced with -3 and -10, do you think the four answers will still be the same?


(b) Investigate, to check your expectation.






Multiplication with integers is associative.

The calculation sequence A can be represented in symbols in only two ways:

3. Express the calculation sequences B, C and D given on page 44 symbolically, without using brackets.





4. Investigate, in the same way that you did for multiplication in question 2, whether addition with integers is associative. Use sequences of four integers.






5. (a) Calculate: 80 - 30 + 40 - 20


(b) Calculate: 80 + (-30) + 40 + (-20)


(c) Calculate: 30 - 80 + 20 - 40


(d) Calculate: (-30) + 80 + (- 20) + 40


(e) Calculate: 20 + 30 - 40 - 80


mixed calculations with integers

1. Calculate.

(a) -3 \times 4 + (-7) \times 9 (b) -20(-4 - 7)

= -12 - 63

= -20(-11)

= -75

= 220

(c) 20 \times (-5) - 30 \times 7 (d) -9(20 - 15)

= -100 - 210

= -9(5)

= –310

= -45

(e) -8 \times (-6) - 8 \times 3 (f) (-26 - 13) \div (-3)

= 48 - 24

= -39 \div (-3)

= 24

= 13

(g) -15 \times (-2) + (-15) \div (-3) (h) -15(2 - 3)

= 30 + 5

= -15(-1)

= 35

= 15

(i) (-5 + -3) \times 7 (j) -5 \times (-3 + 7) + 20 \div (-4)

= -8 \times 7

= -5 \times 4 + (-5)

= –56

= –25

2. Calculate.

(a) 20 \times (-15 + 6) - 5 \times (-2 - 8) - 3 \times (-3 - 8)



(b) 40 \times (7 + 12 - 9) + 25 \div (-5) - 5 \div 5



(c) -50(20 - 25) + 30(-10 + 7) - 20(-16 + 12)




(d) -5 \times (-3 + 12 - 9)



(e) -4 \times (30 - 50) + 7 \times (40 - 70) - 10 \times (60 - 100)



(f) -3 \times (-14 + 6) \times (-13 + 7) \times (-20 + 5)



(g) 20 \times (-5) + 10 \times (-3) + (-5) \times (-6) - (3 \times 5)



(h) -5(-20 - 5) + 10(-7 - 3) - 20(-15 - 5) + 30(-40 - 35)



(i) (-50 + 15 - 75) \div (-11) + (6 - 30 + 12) \div (-6)



80412.png 

2.4 Squares, cubes and roots with integers

squares and cubes of integers

1. Calculate.

(a) 20 \times 20


(b) 20 \times (-20)


2. Write the answers for each of the following:

(a) (-20) \times 20


(b) (-20) \times (-20)


3. Complete the table.

x

1

-1

2

-2

5

-5

10

-10

x2 which isx \times x

1

1

4

4

25

25

100

100

x3

1

-1

8

-8

125

-125

1 000

-1 000

4. In each case, state for which values of x, in the table in question 3, the given statement is true.

(a) x3 is a negative number


(b) x2 is a negative number


(c) x2 > x3


(d) x2 < x3


5. Complete the table.

x

3

-3

4

-4

6

-6

7

-7

x2

9

9

16

16

36

36

49

49

x3

27

-27

64

-64

216

-216

343

-343

6. Ben thinks of a number. He adds 5 to it, and his answer is 12.

(a) What number did he think of?


(b) Is there another number that would also give 12 when 5 is added to it?


7. Lebo also thinks of a number. She multiplies the number by itself and gets 25.

(a) What number did she think of?


(b) Is there more than one number that will give 25 when multiplied by itself?


8. Mary thinks of a number and calculates (the number) \times (the number) \times (the number). Her answer is 27.

What number did Mary think of?


102 is 100 and (-10)2 is also 100.

9. Write the positive square root and the negative square root of each number.

(a) 64


(b) 9


10. Complete the table.

Number

1

4

9

16

25

36

49

64

Positive square root

1

2

3

4

5

6

7

8

Negative square root

-1

-2

-3

-4

-5

-6

-7

-8

11. Complete the tables.

(a)

x

1

2

3

4

5

6

7

8

x3

1

8

27

64

125

216

343

512

(b)

x

-1

-2

-3

-4

-5

-6

-7

-8

x3

-1

-8

-27

-64

-125

-216

-343

-512

33 is 27 and (-5)3 is -125.

3 is called the cube root of 27, because 33 = 27.

-5 is called the cube root of -125 because (-5)3 = -125.

12. Complete the table.

Number

-1

8

-27

-64

-125

-216

1 000

Cube root

-1

2

-3

-4

-5

-6

10

The symbol 79957.png is used to indicate "root".

79952.png represents the cube root of -125. That means 79945.png = -5.

79940.png represents the positive square root of 36, and - 79932.png represents the negative square

root. The "2" that indicates "square" is normally omitted, so 79928.png = 6 and - 79921.png = -6.

13. Complete the table.

79908.png 

79897.png 

79890.png 

- 79885.png

79878.png 

79870.png 

- 79863.png

79856.png 

-2

11

-4

-8

8

-1

-1

-6

1. Use the numbers -8, -5 and -3 to demonstrate each of the following:

(a) Multiplication with integers distributes over addition.

-8(-5 + (-3)) = -8(-5) + (-8)(-3)

(b) Multiplication with integers distributes over subtraction.

-8(-5 - (-3)) = -8(-5) - (-8)(-3)

(c) Multiplication with integers is associative.

-8 \times (-3) \times (-5) = -5 \times (-8) \times (-3)

(d) Addition with integers is associative.

-8 + (-3) + (-5) = -5 + (-8) + (-3)

2. Calculate each of the following without using a calculator:

(a) 5 \times (-2)3 (b) 3 \times (-5)2

=5 \times (-8) = -40

3 \times 25 = 75

(c) 2 \times (-5)3 (d) 10 \times (-3)2

=2 \times (-125) = –250

= 10 \times 9 = 90

3. Use a calculator to calculate each of the following:

(a) 24 \times (-53) + (-27) \times (-34) - (-55) \times 76

= -1 272 + 918 + 4 180 = 3 826

(b) 64 \times (27 - 85) - 29 \times (-47 + 12)

= 64(-58) - 29(-35) = -3 712 + 1 015 = -2 697

4. Use a calculator to calculate each of the following:

(a) -24 \times 53 + 27 \times 34 + 55 \times 76

= 3 826

(b) 64 \times (-58) + 29 \times (47 - 12)

= –2 697

If you don't get the same answers in questions 3 and 4, you have made mistakes.


In this chapter, you will revise work you have done on squares, cubes, square roots and cube roots. You will learn about laws of exponents that will enable you to do calculations using numbers written in exponential form.

Very large numbers are written in scientific notation. Scientific notation is a convenient way of writing very large numbers as a product of a number between 1 and 10 and a power of 10.

3.1 Revision 53

3.2 Working with integers 58

3.3 Laws of exponents 60

3.4 Calculations 70

3.5 Squares, cubes and roots of rational numbers 71

3.6 Scientific notation 74

3 Exponents

3.1 Revision

exponential notation

1. Calculate.

(a) 2 \times 2 \times 2 (b) 2 \times 2 \times 2 \times 2 \times 2 \times 2

= 8

= 64

(c) 3 \times 3 \times 3 (d) 3 \times 3 \times 3 \times 3 \times 3 \times 3

= 27

= 729

Instead of writing 3 \times 3 \times 3 \times 3 \times 3 \times 3 we can write 36.

We read this as "3 to the power of 6". The number 3 is the base, and 6 is the exponent.

When we write 3 \times 3 \times 3 \times 3 \times 3 \times 3 as 36, we are using exponential notation.

2. Write each of the following in exponential form:

(a) 2 \times 2 \times 2 (b) 2 \times 2 \times 2 \times 2 \times 2 \times 2

= 23

= 26

(c) 3 \times 3 \times 3 (d) 3 \times 3 \times 3 \times 3 \times 3 \times 3

= 33

= 36

3. Calculate.

(a) 52 (b) 25

5 \times 5 = 25

2 \times 2 \times 2 \times 2 \times 2 = 32

(c) 102 (d) 152

10 \times 10 = 100

15 \times 15 = 225

(e) 34 (f) 43

3 \times 3 \times 3 \times 3 = 81

4 \times 4 \times 4 = 64

(g) 23 (h) 32

2 \times 2 \times 2 = 8

3 \times 3 = 9

squares

To square a number is to multiply it by itself. The square of 8 is 64 because 8 \times 8 equals 64.

We write 8 \times 8 as 82 in exponential form.

We read 82 as eight squared.

1. Complete the table.

Number

Square the number

Exponential form

Square

(a)

1

1 \times 1

12

1

(b)

2

2 \times 2

22

4

(c)

3

3 \times 3

32

9

(d)

4

4 \times 4

42

16

(e)

5

5 \times 5

52

25

(f)

6

6 \times 6

62

36

(g)

7

7 \times 7

72

49

(h)

8

8 \times 8

82

64

(i)

9

9 \times 9

92

81

(j)

10

10 \times 10

102

100

(k)

11

11 \times 11

112

121

(l)

12

12 \times 12

122

144

2. Calculate the following:

(a) 32 \times 42 (b) 22 \times 32

9 \times 16 = 144

4 \times 9 = 36

(c) 22 \times 52 (d) 22 \times 42

4 \times 25 = 100

4 \times 16 = 64

3. Complete the following statements to make them true:

(a) 32 \times 42 =

12232

(b) 22 \times 32 =

622

(c) 22 \times 52 =

1022

(d) 22 \times 42 =

822

cubes

To cube a number is to multiply it by itself and then by itself again. The cube of 3 is 27 because 3 \times 3 \times 3 equals 27.

We write 3 \times 3 \times 3 as 33 in exponential form.

We read 33 as three cubed.

1. Complete the table.

Number

Cube the number

Exponential form

Cube

(a)

1

1 \times 1 \times 1

13

1

(b)

2

2 \times 2 \times 2

23

8

(c)

3

3 \times 3 \times 3

33

27

(d)

4

4 \times 4 \times 4

43

64

(e)

5

5 \times 5 \times 5

53

125

(f)

6

6 \times 6 \times 6

63

216

(g)

7

7 \times 7 \times 7

73

343

(h)

8

8 \times 8 \times 8

83

512

(i)

9

9 \times 9 \times 9

93

729

(j)

10

10 \times 10 \times 10

103

1 000

2. Calculate the following:

(a) 23 \times 33 (b) 23 \times 53

8 \times 27 = 216

8 \times 125 = 1 000

(c) 23 \times 43 (d) 13 \times 93

8 \times 64 = 512

1 \times 729 = 729

3. Which of the following statements are true? If a statement is false, rewrite it as a true statement.

(a) 23 \times 33 = 63 (b) 23 \times 53 = 73

True

False. 23 \times 53 = 1 000 (which is 103)

(c) 23 \times 43 = 83 (d) 13 \times 93 = 103

True

False. 13 \times 93 = 93 = 729 (and not 103 (1 000))

square and cube roots

To find the square root of a number we ask the question: Which number was multiplied by itself to get a square?

The answer to this question is written as 69178.png = 4.

1. Complete the table.

Number

Square of the number

Square root of the square of the number

Reason

(a)

1

1

1

1 \times 1 = 1

(b)

2

4

2

2 \times 2 = 4

(c)

3

9

3

3 \times 3 = 9

(d)

4

16

4

4 \times 4 = 16

(e)

5

25

5

5 \times 5 = 25

(f)

6

36

6

6 \times 6 = 36

(g)

7

49

7

7 \times 7 = 49

(h)

8

64

8

8 \times 8 = 64

(i)

9

81

9

9 \times 9 = 81

(j)

10

100

10

10 \times 10 = 100

(k)

11

121

11

11 \times 11 = 121

(l)

12

144

12

12 \times 12 = 144

2. Calculate the following. Justify your answer.

(a) 69144.png (b) 69136.png 

= 12, because 12 \times 12 = 144

= 10, because 10 \times 10 = 100

(c) 69081.png (d) 69073.png 

= 9, because 9 \times 9 = 81

= 8, because 8 \times 8 = 64

To find the cube root of a number we ask the question: Which number was multiplied by itself and again by itself to get a cube?

3. Complete the table.

Number

Cube of the number

Cube root of the cube of the number

Reason

(a)

1

1

1

1 \times 1 \times 1 = 1

(b)

2

8

2

2 \times 2 \times 2 = 8

(c)

3

27

3

3 \times 3 \times 3 = 27

(d)

4

64

4

4 \times 4 \times 4 = 64

(e)

5

125

5

5 \times 5 \times 5 = 125

(f)

6

216

6

6 \times 6 \times 6 = 216

(g)

7

343

7

7 \times 7 \times 7 = 343

(h)

8

512

8

8 \times 8 \times 8 = 512

(i)

9

729

9

9 \times 9 \times 9 = 729

(j)

10

1 000

10

10 \times 10 \times 10 = 1 000

4. Calculate the following and give reasons for your answers:

(a) 68977.png (b) 68968.png

= 6, because 6 \times 6 \times 6 = 216

= 2, because 2 \times 2 \times 2 = 8

(c) 68911.png (d) 68902.png 

= 5, because 5 \times 5 \times 5 = 125

= 3, because 3 \times 3 \times 3= 27

(e) 68845.png (f) 68838.png 

= 4, because 4 \times 4 \times 4 = 64

= 10, because 10 \times 10 \times 10 = 1 000

3.2 Working with integers

representing integers in exponential form

1. Calculate the following, without using a calculator:

(a) -2 \times -2 \times -2 (b) -2 \times -2 \times -2 \times -2

= -8

= 16

(c) -5 \times -5 (d) -5 \times -5 \times -5

= 25

= -125

(e) -1 \times -1 \times -1 \times -1 (f) -1 \times -1 \times -1

= 1

= -1

2. Calculate the following:

(a) -22 (b) (-2)2

= -4

= -2 \times -2 = 4

(c) (-5)3 (d) -53

= -5 \times -5 \times -5 = -125

= -125

3. Use your calculator to calculate the answers to question 2.

(a) Are your answers to question 2(a) and (b) different or the same as those of the calculator?


(b) If your answers are different to those of the calculator, try to explain how the calculator did the calculations differently from you.





The calculator "understands" -52 and (-5)2 as two different numbers.

(-5)2 as -5 \times -5 = 25

4. Write the following in exponential form:

(a) -2 \times -2 \times -2 (b) -2 \times -2 \times -2 \times -2

= (-2)3

= (-2)4

(c) -5 \times -5 (d) -5 \times -5 \times -5

= (-5)2

= (-5)3

(e) -1 \times -1 \times -1 \times -1 (f) -1 \times -1 \times -1

= (-1)4

= (-1)3

5. Calculate the following:

(a) (-3)2 (b) (-3)3

= -3 \times -3

= -3 \times -3 \times -3

= 9

= -27

(c) (-2)4 (d) (-2)6

= -2 \times -2 \times -2 \times –2

= -2 \times -2 \times -2 \times -2 \times -2 \times -2

= 16

= 64

(e) (-2)5 (f) (-3)4

= -2 \times -2 \times -2 \times -2 \times -2

= -3 \times -3 \times -3 \times -3

= -32

= 81

6. Say whether the sign of the answer is negative or positive. Explain why.

(a) (-3)6 (b) (-5)11

Positive. The power is even.

Negative. The power is odd

(c) (-4)20 (d) (-7)5

Positive. The power is even.

Negative. The power is odd.

7. Say whether the following statements are true or false. If a statement is false rewrite it as a correct statement.

(a) (-3)2 = -9 (b) -32 = 9

False. (-3)2 = -3 \times -3 = 9

False. -32 = -9 because -32 = -(32)

(c) (-52) = -52 (d) (-1)3 = -13

True. Both are equal to –25.

True. Both are equal to -1.

(e) (-6)3 = -18 (f) (-2)6 = 26

False. -6 \times -6 \times -6 = -216

True. Both are equal to 64.

3.3 Laws of exponents

product of powers

1. A product of 2s is given below. Describe it using exponential notation, that is, write it as a power of 2.

2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2


2. Express each of the following as a product of the powers of 2, as indicated by the brackets.

(a) (2 \times 2 \times 2) \times (2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2)


(b) (2 \times 2 \times 2 \times 2 \times 2) \times (2 \times 2 \times 2 \times 2 \times 2) \times (2 \times 2)


(c) (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (2 \times 2)


(d) (2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (2 \times 2 \times 2)


(e) (2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2) \times (2 \times 2)


3. Complete the following statements so that they are true. You may want to refer to your answers to question 2 (a) to (e) to help you.

(a) 23 \times


= 212

(b) 25 \times


\times 22 =212

(c) 22 \times 22 \times 22 \times 22 \times 22 \times 22 =


(d) 28 \times


= 212

(e) 23 \times 23 \times 23 \times


= 212

(f) 26 \times


= 212

(g) 22\times 210 =


Suppose we are asked to simplify: 32 \times 34.

The solution is: 32 \times 34 = 9 \times 81

= 729

= 36

The base (3) is a repeated factor. The exponents (2 and 4) tell us the number of times each factor is repeated.

We can explain this solution in the following manner:

32 \times 34 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 36

67647.png
67654.png
67663.png

2 factors 4 factors 6 factors

4. Complete the table.

Product of powers

Repeated factor

Total number of times the factor is repeated

Simplified form

(a)

27 \times 23

2

10

210

(b)

52\times 54

5

6

56

(c)

41\times 45

4

6

46

(d)

63\times 62

6

5

65

(e)

28 \times 22

2

10

210

(f)

53 \times 53

5

6

56

(g)

42 \times 44

4

6

46

(h)

21 \times 29

2

10

210

When you multiply two or more powers that have the same base, the answer has the same base, but its exponent is equal to the sum of the exponents of the numbers you are multiplying.

5. What is wrong with these statements? Correct each one.

(a) 23 \times 24 = 212 (b) 10 \times 102 \times 103 = 101 \times 2 \times 3 = 106

= 23+ 4 = 27

= 101+ 2 + 3 = 106

(c) 32 \times 33 = 36 (d) 53 \times 52 = 15 \times 10

= 32+ 3 = 35

= 53+ 2 = 55

6. Express each of the following numbers as a single power of 10.

Example: 1 000 000 as a power of 10 is 106.

(a) 100 (b) 1 000 (c) 10 000

= 102


= 103


= 104


(d) 102 \times 103 \times 104 (e) 100 \times 1 000 \times 10 000 (f) 1 000 000 000

= 102 + 3 + 4


= 102 \times 103 \times 104


= 109


= 109


= 109


7. Write each of the following products in exponential form:

(a) x \times x \times x \times x \times x \times x \times x \times x \times x =


(b) (x \times x) \times (x \times x \times x) \times (x \times x \times x \times x)

= x2 \times x3 \times x4 = x2 + 3 + 4 = x9

(c) (x \times x \times x \times x) \times (x \times x) \times (x \times x) \times x

= x4 \times x2 \times x2 \times x = x4 + 2 + 2 + 1 = x9

(d) (x \times x \times x \times x \times x \times x) \times (x \times x \times x)


(e) (x \times x \times x) \times (y \times y \times y)


(f) (a \times a) \times (b \times b)


8. Complete the table.

Product of powers

Repeated factor

Total number of times the factor is repeated

Simplified form

(a)

x7 \times x3

x

10

x10

(b)

x2 \times x4

x

6

x6

(c)

x1 \times x5

x

6

x6

(d)

x3 \times x2

x

5

x5

(e)

x8 \times x2

x

10

x10

(f)

x3 \times x3

x

6

x6

(g)

x1 \times x9

x

10

x10

raising a power to a power

1. Complete the table of powers of 2.

x

1

2

3

4

5

6

7

8

9

10

11

2x

2

4

8

16

32

64

128

256

512

1 024

2 048

21

22

23

24

25

26

27

28

29

210

211

x

12

13

14

15

16

17

18

2x

4 096

8 192

16 384

32 768

65 536

131 072

262 144

212

213

214

215

216

217

218

2. Complete the table of powers of 3.

x

1

2

3

4

5

6

7

8

9

3x

3

9

27

81

243

729

2 187

6 561

19 683

31

32

33

34

35

36

37

38

39

x

10

11

12

13

14

3x

59 049

177 147

531 441

1 594 323

4 782 969

310

311

312

313

314

3. Complete the table. You can read the values from the tables you made in questions 1 and 2.

Product of powers

Repeated factor

Power of powernotation

Total number of repetitions

Simplified form

Value

24 \times 24 \times 24

2

(24)3

12

212

4 096

32 \times 32 \times 32 \times 32

3

(32)4

8

38

6 561

23 \times 23 \times 23 \times 23 \times 23

2

(23)5

15

215

32 768

34 \times 34 \times 34

3

(34)3

12

312

531 441

26 \times 26 \times 26

2

(26)3

18

218

262 144

4. Use your table of powers of 2 to find the answers for the following:

(a) 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 =


=


(b) (2 \times 2 \times 2 \times 2) \times (2 \times 2 \times 2 \times 2) \times (2 \times 2 \times 2 \times 2) =


=


(c) 163 =


=


=


5. Use your table of powers of 2 to find the answers for the following:

(a) Is 163 = 212?


(b) Is 24 \times 24 \times 24 = 212


(c) Is 24 \times 23 = 212?


(d) Is (24)3 = 24 \times 24 \times 24?


(e) Is (24)3 = 212?


(f) Is (24)3 = 24 + 3?


(g) Is (24)3 = 24 \times 3?


(h) Is (22)5 = 22 + 5?


6. (a) Express 85 as a power of 2. It may help to first express 8 as a power of 2.


(b) Can (23) \times (23) \times (23) \times (23) \times (23) be expressed as (23)5?


(c) Is (23)5 = 23 + 5 or is (23)5 = 23 \times 5?


7. (a) Express 43 as a power of 2.


(b) Calculate 22 \times 22 \times 22 and express your answer as a single power of 2.


(c) Can (22) \times (22) \times (22) be expressed as (22)3?


(d) Is (22)3 = 22 + 3 or is (22)3 = 22 \times 3?


8. Simplify the following.

Example: (102)2 = 102 \times 102 = 102 + 2 = 104 = 10 000

(a) (33)2


(b) (43)2


(c) (24)2


(d) (92)2


(e) (33)3


(f) (43)3


(g) (54)3


(h) (92)3


(am)n = am \times n, where m and n are natural numbers and a is not equal to zero.

9. Simplify.

(a) (54)10


(b) (104)5


(c) (64)4


(d) (54)10


10. Write 512 as a power of powers of 5 in two different ways.


To simplify (x2)5 we can write it out as a product of powers or we can use a shortcut.

(x2)5 = x2 \times x2 \times x2 \times x2 \times x2

= x \times x \times x \times x \times x \times x \times x \times x \times x \times x = x10

100389.png
100400.png
100404.png
100408.png
100413.png

2 \times 5 factors = 10 factors

11. Complete the table.

Expression

Write as a product of the powers and then simplify

Use the rule (am)n to simplify

(a)

(a4)5

a4 \times a4 \times a4 \times a4 \timesa4

= a4 + 4 + 4 + 4 + 4 = a20

(a4)5 = a4 \times 5 = a20

(b)

(b10)5

b10 \times b10 \times b10 \times b10 \times b10

= b10 + 10 + 10 + 10 +10

= b50

(b10)5 = b10 \times 5

= b50

(c)

(x7)3

x7 \times x7 \times x7

= x7 + 7 + 7

= x21

(x7)3 = x7 \times 3

= x21

(d)

(s6)4

s6 \times s6 \times s6 \times s6

= s6 + 6 + 6 + 6

= s24

(s6)4 = s6 \times 4

= s24

(e)

(y3)7

y3 \times y3 \times y3 \times y3 \times y3 \times y3 \times y3

= y3+ 3 + 3 + 3 + 3 + 3 + 3 + 3

= y21

y3 \times 7 = y21

power of a product

1. Complete the table. You may use your calculator when you are not sure of a value.

x

1

2

3

4

5

(a)

2x

21 = 2

22 = 4

23 = 8

24 = 16

25 = 32

(b)

3x

31 = 3

32 = 9

33 = 27

34 = 81

35 = 243

(c)

6x

61 = 6

62 = 36

63 = 216

64 = 1 296

65 = 6 480

2. Use the table in question 1 to answer the questions below. Are these statements true or false? If a statement is false rewrite it as a correct statement.

(a) 62 = 22 \times 32 (b) 63 = 23 \times 33

True. 36 = 4 \times 9

True. 216 = 8 \times 27

(c) 65 = 25 \times 35 (d) 68 = 24 \times 34

True. 6 480 = 32 \times 243

False. 24 \times 34 = 64 not 68

3. Complete the table.

Expression

The bases of the expression are factors of …

Equivalent expression

(a)

26 \times 56

10

106

(b)

32 \times 42

12

122

(c)

42 \times 22

8

82

(d)

75 \times 85

56

565

(e)

23 \times 153

30

303

(f)

35 \times x5

3x

(3x)5

(g)

72 \times z2

7z

(7z)2

(h)

43 \times y3

4y

(4y)3

(i)

2m

(2m)6

(j)

2m

(2m)3

(k)

210 \times y10

2y

(2y)10

122 can be written in terms of its factors as (2 \times 6)2 or as (3 \times 4)2.

We already know that 122 = 144.

What this tells us is that both (2 \times 6)2 and (3 \times 4)2 also equal 144.

We write 122 = (2 \times 6)2 or 122 = (3 \times 4)2

= 22 \times 62 = 32 \times 42

= 4 \times 36 = 9 \times 16

= 144 = 144

Aproduct raised to a power is the product of the factors each raised to the same power.

4. Write each of the following expressions as an expression with one base:

Example: 310 \times 210 = (3 \times 2)10 = 610

(a) 32 \times 52 (b) 53 \times 23 (c) 74 \times 44

= (3 \times 5)2


= (5 \times 2)3


= (7 \times 4)4


= 152


= 103


= 284


(d) 23 \times 63 (e) 44 \times 24 (f) 52 \times 72

= (2 \times 6)3


= (4 \times 2)4


= (5 \times 7)2


= 123


= 84


= 352


5. Write the following as a product of powers:

Example: (3x)3 = 33 \times x3 = 27x3

(a) 63 (b) 152 (c) 214

= (2 \times 3)3


= (3 \times 5)2


= (3 \times 7)4


= 23 \times 33


= 32 \times 52


= 34 \times 74



(d) 65 (e) 182 (f) (st)7

= (2 \times 3)5


= (2 \times 9)2


= s7t7


= 25 \times 35


= 22 \times 92


(g) (ab)3 (h) (2x)2 (i) (3y)5

= a3b3


= 22x2


= 35y5


= 4x2


= 243y5


(j) (3c)2 (k) (gh)4 (l) (4x)3

= 32c2


= g4h4


= 43x3


= 9c2


= 64x3


6. Simplify the following expressions:

Example: 32 \times m2 = 9 \times m2 = 9m2

(a) 35 \times b5 (b) 26 \times y6 (c) x2 \times y2

= 243b5


= 64y6


= x2y2


(d) 104 \times x4 (e) 33 \times x3 (f) 52 \times t2

= 10 000x4


= 27x3


= 25t2


(g) 63 \times m7 (h) 122 \times a2 (i) n3 \times p9

= 216m7


= 144a2


= n3p9


a quotient of powers

Consider the following table:

x

1

2

3

4

5

6

2x

2

4

8

16

32

64

3x

3

9

27

81

243

729

5x

5

25

125

625

3 125

15 625

Answer questions 1 to 4 by referring to the table when you need to.

1. Give the value of each of the following:

(a) 34 (b) 25 (c) 56







2. (a) Calculate 36 \div 33 (Read the values of 36 and 33 from the table and then divide.

You may use a calculator where necessary.)

To calculate 45– 3 we first do the calculation in the exponent, that is, we subtract 3 from 5. Then we can calculate 42 as 4 \times 4 = 16.


(b) Calculate 36- 3


(c) Is 36 \div 33 equal to 33? Explain.


3. (a) Calculate the value of 26- 2 (b) Calculate the value of 26 \div 22

26- 2 = 24 = 16

26 \div 22 = 64 \div 4 = 16

(c) Calculate the value of 26\div 2 (d) Read from the table the value of 23

26\div 2 = 23 = 8

8

(e) Read from the table the value of 24


(f) Which of the statements below is true? Give an explanation for your answer.

A. 26 \div 22 = 26 - 2 = 24 B. 26 \div 22 = 26 \div 2 = 23



4. Say which of the statements below are true and which are false. If a statement is false rewrite it as a correct statement.

(a) 56 \div 54 = 56 \div 4 (b) 34 - 1 = 34 \div 3

False. 56 \div 54 = 15 625 \div 625 = 25

True. 34- 1 = 33 = 27

56- 4 = 52

34 \div 3 = 81 \div 3 = 27

(c) 56 \div 5 = 56 - 1 (d) 25 \div 23 = 22

True

True

am \div an = am – n where m and n are natural numbers and m is a number greater than n and a is not zero.

5. Simplify the following. Do not use a calculator.

Example: 317 \div 312 = 317 - 12 = 35 = 243

(a) 212 \div 210 (b) 617 \div 614

= 212 - 10

= 617 - 14

= 22

= 63

= 4

= 216

(c) 1020 \div 1014 (d) 511 \div 58

= 1020 – 14

= 511 - 8

= 106

= 53

= 1 000 000

= 125

6. Simplify:

(a) x12 \div x10 (b) y17 \div y14

= x12 - 10

= y17 - 14

= x2

= y3

(c) t20 \div t14 (d) n11 \div n8

= t20 - 14

= n11 - 8

= t6

= n3

the power of zero

1. Simplify the following:

(a) 212 \div 212 (b) 617 \div 617

= 212 - 12

= 617 - 17

= 20

= 60

(c) 614 \div 614 (d) 210 \div 210

= 614 -14

= 210 - 10

= 60

= 20

We define a0 = 1.

Any number raised to the power of zero is always equal to 1.

2. Simplify the following:

(a) 1000 (b) x0 (c) (100x)0 (d) (5x3)0





64340.png 

3.4 Calculations

mixed operations

Simplify the following:

1. 33 + 64330.png \times 2 2. 5 \times (2 + 3)2 + (-1)0


= 5 \times 52 + 1


= 125 + 1


= 126

3. 32 \times 23 + 5 \times 64178.png 4. 64170.png + (4 - 1)2

= 9 \times 8 + 5 \times 10

= 70583.jpg + 32


= 1 + 9


= 10

5. 64019.png \times 64012.png + 64005.png + 32 \times 10 6. 43 \div 23 + 63996.png


= 64 \div 8 + 12


= 8 + 12


= 20

63849.png 

3.5 Squares, cubes and roots of rational numbers

squaring a fraction

Squaring or cubing a fraction or a decimal fraction is no different from squaring or cubing an integer.

1. Complete the table.

Fraction

Square the fraction

Value of the square of the fraction

(a)

63831.png 

63822.png \times 63817.png 

63812.png \times 63803.png = 63795.png 

(b)

63787.png 

(c)

63761.png 

(d)

63725.png 

(e)

63696.png 

(f)

63669.png 

(g)

63635.png 

(h)

95462.png 

2. Calculate the following:

(a) 63599.png (b) 63591.png (c) 63584.png 

= 70537.jpg \times 70527.jpg = 70519.jpg

= 70561.jpg \times 70553.jpg = 70542.jpg

= 70579.jpg \times 70572.jpg = 70565.jpg

3. (a) Use the fact that 0,6 can be written as 63504.png to calculate (0,6)2.

(0,6)2 = 63492.jpg = 63484.jpg = 0,36

(b) Use the fact that 0,8 can be written as 63476.png to calculate (0,8)2.

(0,8)2 = 63467.jpg = 63459.jpg = 0,64

finding the square root of a fraction

1. Complete the table.

Fraction

Writing the fraction as a product of factors

Square root

(a)

63451.png 

(b)

63419.png 

(c)

63382.png 

(d)

63351.png 

2. Determine the following:

(a) 63303.png (b) 63296.png (c) 63287.png (d) 63280.png 

= 70489.jpg 

= 70496.jpg 

= 70503.jpg 

= 70511.jpg 

3. (a) Use the fact that 0,01 can be written as 63178.png to calculate 63170.png.

= =

(b) Use the fact that 0,49 can be written as 63133.png to calculate 63124.png.

= = = 0,7

4. Calculate the following:

(a) 63092.png (b) 63085.png (c) 63076.png

= 70444.jpg = 70435.jpg = 0,3

= 70467.jpg = 70454.jpg = 0,8

= 70481.jpg = 70474.jpg = 1,2

cubing a fraction

One half cubed is equal to one eighth.

We write this as 62996.png = 62992.png \times 62983.png \times 62975.png = 62966.png

1. Calculate the following:

(a) 62949.png (b) 62941.png (c) 62934.png (d) 62927.png

= 70377.jpg \times 70371.jpg \times 70363.jpg 

= 70399.jpg \times 70395.jpg \times 70383.jpg 

= 70414.jpg \times 70410.jpg \times 70403.jpg 

= 70430.jpg \times 70425.jpg \times 70418.jpg 

= 70333.jpg 

= 70341.jpg 

= 70348.jpg 

= 70356.jpg 

2. (a) Use the fact that 0,6 can be written as 62729.png to find (0,6)3.

(0,6)3 = = = 0,216

(b) Use the fact that 0,8 can be written as 62706.png to calculate (0,8)3.

(0,8)3 = = = 0,512

(c) Use the fact that 0,7 can be written as 95481.png to calculate (0,7)3.

(0,8)3 = = = 0,512

3.6 Scientific notation

very large numbers

1. Express each of the following as a single number. Do not use a calculator.

Example: 7,56 \times 100 can be written as 756.

(a) 3,45 \times 100 (b) 3,45 \times 10 (c) 3,45 \times 1 000







(d) 2,34 \times 102 (e) 2,34 \times 10 (f) 2,34 \times 103







(g) 104 \times 102 (h) 100 \times 106 (i) 3,4 \times 105

106 = 1 000 000








We can write 136 000 000 as 1,36 \times 108.

1,36 \times 108 is called the scientific notation for 136 000 000.

2. Write the following numbers in scientific notation:

(a) 367 000 000 (b) 21 900 000


2,19 \times 107

(c) 600 000 000 000 (d) 178


1,78 \times 102

3. Write each of the following numbers in the ordinary way.

For example: 3,4 \times 105 written in the ordinary way is 340 000.

(a) 1,24 \times 108 (b) 9,2074 \times 104


92 074

(c) 1,04 \times 106 (d) 2,05 \times 103


2 050

4. The age of the universe is 15 000 000 000 years. Express the age of the universe in scientific notation.


5. The average distance from the Earth to the Sun is 149 600 000 km. Express this distance in scientific notation.


Because it is easier to multiply powers of ten without a calculator, scientific notation makes it possible to do calculations in your head.

6. Explain why the number 24 \times 103 is not in scientific notation.


7. Calculate the following. Do not use a calculator.

Example: 3 000 000 \times 90 000 000 = 3 \times 106 \times 9 \times 107 = 3 \times 9 \times 106 + 7

= 27 \times 1013 = 270 000 000 000 000

(a) 13 000 \times 150 000 (b) 200 \times 6 000 000


= 2 \times 102 \times 6 \times 106


= 2 \times 6 \times 102 \times 106


= 2 \times 6 \times 102+ 6


= 12 \times 108


= 1 200 000 000

(c) 120 000 \times 120 000 000 (d) 2,5 \times 40 000 000


= 2,5 \times 4 \times107


= 10 \times 107


= 108


= 100 000 000


8. Use > or < to compare these numbers:

(a) 1,3 \times 109

2,4 \times 107 (b) 6,9 \times 102

4,5 \times 103

(c) 7,3 \times 104

7,3 \times 102 (d) 3,9 \times 106

3,7 \times 107

1. Calculate:

(a) 112

= 11 \times 11 = 121

(b) 32 \times 42

= 9 \times 16 = 144

(c) 63

= 6 \times 6 \times 6 = 216

(d) 61625.png

= 11

(e) (-3)2

= -3 \times -3 = 9

(f) 61567.png

= 5

2. Simplify:

(a) 34 \times m6

= 81m6

(b) b2 \times n6

= b2n6

(c) y12 \div y5

= y12 - 5 = y7

(d) (102)3

= 106

(e) (2w2)3

= 23 \times w6 = 8w6

(f) (3d5)(2d)3

= 3a5 \times 8a3 = 24a8

3. Calculate:

(a) 61391.png

= 70324.jpg = 70317.jpg 

(b) 61360.png

= 70309.jpg = 70297.jpg 

(c) (64y 2)0

= 1

(d) (0,7)2

= 0,49

4. Simplify:

(a) (22 + 4)2 + 97796.png (b) 97788.png - 5 \times 32

=(4 + 4)2 + 97864.jpg or (4 + 4)2 + 97856.jpg

= -5 - 5 \times 9

= 82 + 22 = 82 + 4

= -5 - 45

= 64 + 4 = 68 =64 + 4 = 68

= -50

5. Write 3 \times 109 in the ordinary way.

3 000 000 000

6. The first birds appeared on Earth about 208 000 000 years ago. Write this number in scientific notation.

2,08 \times 106

gr8ch3.tif

In this chapter you will learn to create, recognise, describe, extend and make generalisations about numeric and geometric patterns. Patterns allow us to make predictions. You will also work with different representations of patterns, such as flow diagrams and tables.

4.1 The term–term relationship in a sequence 79

4.2 The position–term relationship in a sequence 84

4.3 Investigate and extend geometric patterns 86

4.4 Describe patterns in different ways 90

MathsA%20Grade8%20Chapter%204%20TG_final-2.tif

4 Numeric and geometric patterns

4.1 The term–term relationship in a sequence

going from one term to the next

Write down the next three numbers in each of the sequences below. Also explain in writing, in each case, how you figured out what the numbers should be.

A list of numbers which form a pattern is called a sequence. Each number in a sequence is called a term of the sequence. The first number is the first term of the sequence.

1. Sequence A: 2; 5; 8; 11; 14; 17; 20; 23;



2. Sequence B: 4; 5; 8; 13; 20; 29; 40;



3. Sequence C: 1; 2; 4; 8; 16; 32; 64;



4. Sequence D: 3; 5; 7; 9; 11; 13; 15; 17; 19;



5. Sequence E: 4; 5; 7; 10; 14; 19; 25; 32; 40;



6. Sequence F: 2; 6; 18; 54; 162; 486;



7. Sequence G: 1; 5; 9; 13; 17; 21; 25; 29; 33;



8. Sequence H: 2; 4; 8; 16; 32; 64;



Numbers that follow one another are said to be consecutive.

adding or subtracting the same number

1. Which sequences on the previous page are of the same kind as sequence A? Explain your answer.



Amanda explains how she figured out how to continue sequence A:

I looked at the first two numbers in the sequence and saw that I needed 3 to go from 2 to 5. I looked further and saw that I also needed 3 to go from 5 to 8. I tested that and it worked for all the next numbers.

This gave mearule I could use to extend the sequence: add 3 to each number to find the next number in the pattern.

Tamara says you can also find the pattern by working backwards and subtracting 3 each time:

When the differences between consecutive terms of a sequence are the same, we say the difference is constant.

14 - 3 = 11; 11 - 3 = 8; 8 - 3 = 5; 5 - 3 = 2

2. Provide a rule to describe the relationship between the numbers in the sequence. Use this rule to calculate the missing numbers in the sequence.

(a) 1; 8; 15;


;


;


;


;


; ...


(b) 10 020;


;


;


; 9 980; 9 970;


;


; 9 940; 9 930; ...


(c) 1,5; 3,0; 4,5;


;


;


;


;


; ...


(d) 2,2; 4,0; 5,8;


;


;


;


;


; ...


(e) 45 98129.png; 46 98119.png; 47 98112.png; 48;


;


;


;


;


; …


(f)


; 100,49; 100,38; 100,27;


;


; 99,94; 99,83; 99,72; …



3. Complete the table below.

Input number

1

2

3

4

5

8

12

23

n

Input number + 7

8

9

10

11

12

15

19

30

n + 7

multiplying or dividing with the same number

Take another look at sequence F: 2; 6; 18; 54; 162; 486; ...

Piet explains that he figured out how to continue the sequence F:

Ilooked at the first two terms in the sequence and wrote 2 \times ? = 6.

When I multiplied the first number by 3, I got the second number: 2 \times 3 = 6.

Ithen checked to see if I could find the next number if I multiplied 6 by 3: 6 \times 3 = 18.

Icontinued checking in this way: 18 \times 3 = 54; 54 \times 3 = 162 and so on.

This gave me arule I can use to extend the sequence and my rule was: multiply each number by 3 to calculate the next number in the sequence.

Zinhle says you can also find the pattern by working backwards and dividing by 3 each time:

54 \div 3 = 18; 18 \div 3 = 6; 6 \div 3 = 2

The number that we multiply with to get the next term in the sequence is called a ratio. If the number we multiply with remains the same throughout the sequence, we say it is a constant ratio.

1. Check whether Piet's reasoning works for sequence H: 2; 4; 8; 16; 32; 64; ...


2. Describe, in words, the rule for finding the next number in the sequence. Also write down the next five terms of the sequence if the pattern is continued.

(a) 1; 10; 100; 1 000;



(b) 16; 8; 4; 2;



(c) 7; -21; 63; -189;



(d) 3; 12, 48;



(e) 2 187; -729; 243; -81;



3. (a) Fill in the missing output and input numbers:

72618.png 

(b) Complete the table below:

What is the term-to-term rule for the output numbers here, + 6 or \times 6?

Input numbers

1

2

3

4

5

6

12

x

Output numbers

6

12

18

24

30

36

72

neither adding nor multiplying by the same number

1. Consider sequences A to H again and answer the questions that follow:

Sequence A: 2; 5; 8; 11; 14; 17; 20; 23; ...

Sequence B: 4; 5; 8; 13; 20; 29; 40; …

Sequence C: 1; 2; 4; 8; 16; 32; 64; …

Sequence D: 3; 5; 7; 9; 11; 13; 15; 17; 19; ...

Sequence E: 4; 5; 7; 10; 14; 19; 25; 32; 40; …

Sequence F: 2; 6; 18; 54; 162; 486; …

Sequence G: 1; 5; 9; 13; 17; 21; 25; 29; 33; ...

Sequence H: 2; 4; 8; 16; 32; 64; ….

(a) Which other sequence(s) is/are of the same kind as sequence B? Explain.


(b) In what way are sequences B and E different from the other sequences?



There are sequences where there is neither a constant difference nor a constant ratio between consecutive terms and yet a pattern still exists, as in the case of sequences B and E.

2. Consider the sequence: 10; 17; 26; 37; 50; ...

(a) Write down the next five numbers in the sequence.


(b) Eric observed that he can calculate the next term in the sequence as follows: 10 + 7 = 17; 17 + 9 = 26; 26 + 11 = 37. Use Eric's method to check whether your numbers in question (a) above are correct.




3. Which of the statements below can Eric use to describe the relationship between the numbers in the sequence in question 2? Test the rule for the first three terms of the sequence and then simply write "yes" or "no" next to each statement.

(a) Increase the difference between consecutive terms by 2 each time





(b) Increase the difference between consecutive terms by 1 each time





(c) Add two more than you added to get the previous term






4. Provide a rule to describe the relationship between the numbers in the sequences below. Use your rule to provide the next five numbers in the sequence.

(a) 1; 4; 9; 16; 25;



(b) 2; 13; 26; 41; 58;



(c) 4; 14; 29; 49; 74;



(d) 5; 6; 8; 11; 15; 20;



4.2 The position–term relationship in a sequence

using position to make predictions

1. Take another look at sequences A to H. Which sequence(s) are of the same kind as sequence A? Explain.

Sequence A: 2; 5; 8; 11; 14; 17; 20; 23;...

Sequence B: 4; 5; 8; 13; 20; 29; 40; …

Sequence C: 1; 2; 4; 8; 16; 32; 64; …

Sequence D: 3; 5; 7; 9; 11; 13; 15; 17; 19; ...

Sequence E: 4; 5; 7; 10; 14; 19; 25; 32; 40; …

Sequence F: 2; 6; 18; 54; 162; 486; …

Sequence G: 1; 5; 9; 13; 17; 21; 25; 29; 33; ...

Sequence H: 2; 4; 8; 16; 32; 64; …


Sizwe has been thinking about Amanda and Tamara's explanations of how they worked out the rule for sequence A and has drawn up a table. He agrees with them but says that there is another rule that will also work. He explains:

My table shows the terms in the sequence and the difference between consecutive terms:

1st term

2nd term

3rd term

4th term

A:

5

8

11

14

differences

+3

+3

+3

+3

+3

+3

+3

+3

+3

Sizwe reasons that the following rule will also work:

Multiply the position of the number by 3 and add 2 to the answer.

I can write this rule as a number sentence: Position of the number \times 3 + 2

I use my number sentence to check: 1\times 3 + 2 = 5; 2 \times 3 + 2 = 8; 3 \times 3 + 2 = 11

2. (a) What do the numbers in bold in Sizwe's number sentence stand for?


(b) What does the number 3 in Sizwe's number sentence stand for?


3. Consider the sequence 5; 8; 11; 14; ...

Apply Sizwe's rule to the sequence and determine:

(a) term number 7 of the sequence (b) term number 10 of the sequence

7 \times 3 + 2 = 21 + 2 = 23



(c) the 100th term of the sequence


4. Consider the sequence: 3; 5; 7; 9; 11; 13; 15; 17; 19;..

(a) Use Sizwe's explanation to find a rule for this sequence.


(b) Determine the 28th term of the sequence.



more predictions

Complete the tables below by calculating the missing terms.

1.

Position in sequence

1

2

3

4

10

54

Term

4

7

10

13

31

163





2.

Position in sequence

1

2

3

4

8

16

Term

4

9

14

19

39

79





3.

Position in sequence

1

2

3

4

7

30

Term

3

15

27

39

75

351





4. Use the rule Position in the sequence \times (position in the sequence + 1) to complete the table below.

Position in sequence

1

2

3

4

5

6

Term

2

6

12

20

30

42

72281.png 

4.3 Investigating and extending geometric patterns

square numbers

A factory makes window frames. Type 1 has one windowpane, type 2 has four windowpanes, type 3 has nine windowpanes, and so on.

Type 1 Type 2 Type 3 Type 4

72249.png

1. How many windowpanes will there be in type 5?


2. How many windowpanes will there be in type 6?


3. How many windowpanes will there be in type 7?


4. How many windowpanes will there be in type 12? Explain.



5. Complete the table. Show your calculations.

Frame type

1

2

3

4

15

20

Number of windowpanes

1

4

9

16

225

400


The symbol n is used below to represent the position number in the expression that gives the rule (n2) when generalising.

In algebra we think of a square as a number that is obtained by multiplying a number by itself. So 1 is also a square because 1 \times 1 = 1.

72098.png

12 22 32 42 52 ... n2

triangular numbers

Therese uses circles to form a pattern of triangular shapes:

72088.png 

1. If the pattern is continued, how many circles must Therese have

(a) in the bottom row of picture 5?


(b) in the second row from the bottom of picture 5?


(c) in the third row from the bottom of picture 5?


(d) in the second row from the top of picture 5?


(e) in the top row of picture 5?


(f) in total in picture 5? Show your calculation.



2. How many circles does Therese need to form triangle picture 7? Show the calculation.


3. How many circles does Therese need to form triangle picture 8?


4. Complete the table below. Show all your work.

Picture number

1

2

3

4

5

6

12

15

Number of circles

1

3

6

10

15

21

78

120





More than 2 500 years ago, Greek mathematicians already knew that the numbers 3, 6, 10, 15 and so on could form a triangular pattern. They represented these numbers with dots which they arranged in such a way that they formed equilateral triangles, hence the name triangular numbers. Algebraically we think of them as sums of consecutive natural numbers starting with 1.

Let us revisit the activity on triangular numbers that we did on the previous page.

71938.png 

So far, we have determined the number of circles in the pattern by adding consecutive natural numbers. If we were asked to determine the number of circles in picture 200, for example, it would take us a very long time to do so. We need to find a quicker method of finding any triangular number in the sequence.

Consider the arrangement below.

71929.png 

We have added the yellow circles to the original blue circles and then rearranged the circles in such a way that they are in a rectangular form.

5. Picture 2 is 3 circles long and 2 circles wide. Complete the following sentences:

(a) Picture 3 is


circles long and


circles wide.

(b) Picture 1 is


circles long and


circle wide.

(c) Picture 4 is


circles long and


circles wide.

(d) Picture 5 is


circles long and


circles wide.

6. How many circles will there be in a picture that is:

(a) 10 circles long and 9 circles wide?


(b) 7 circles long and 6 circles wide?


(c) 6 circles long and 5 circles wide?


(d) 20 circles long and 19 circles wide?


Suppose we want to have a quicker method to determine the number of circles in picture 15. We know that picture 15 is 16 circles long and 15 circles wide. This gives a total of 15 \times 16 = 240 circles. But we must compensate for the fact that the yellow circles were originally not there by halving the total number of circles. In other words, the original figure has 240 \div 2 = 120 circles.

7. Use the above reasoning to calculate the number of circles in:

(a) picture 20

Circles in picture 20 = = = 210


(b) picture 35

Circles in picture 35 = = = 630


4.4 Describing patterns in different ways

t-shaped numbers…

The pattern below is made from squares.

71576.png

1 2 3 4

1. (a) How many squares will there be in pattern 5?


(b) How many squares will there be in pattern 15?


(c) Complete the table.

Pattern number

1

2

3

4

5

6

20

Number of squares

1

4

7

10

13

16

58

Below are three different methods or plans to calculate the number of squares for pattern 20. Study each one carefully.

Plan A:

To get from 1 square to 4 squares, you have to add 3 squares. To get from 4 squares to 7 squares, you have to add 3 squares. To get from 7 squares to 10 squares, you have to add 3 squares. So continue to add 3 squares for each pattern until pattern 20.

Plan B:

Multiply the pattern number by 3, and subtract 2. So pattern 20 will have 20 \times 3 - 2 squares.

Plan C:

The number of squares in pattern 5 is 13. So pattern 20 will have 13 \times 4 = 52 squares because 20 = 5 \times 4.

2. (a) Which method or plan (A, B or C) will give the right answer? Explain why.


(b) Which of the above plans did you use? Explain why?



(c) Can this flow diagram be used to calculate the number of squares?


71484.png  

… and some other shapes

1. Three figures are given below. Draw the next figure in the tile pattern.

71475.png 

2. (a) If the pattern is continued, how many tiles will there be in the 17th figure? Answer this question by analysing what happens.




(b) Thato decides that it easier for him to see the pattern when the tiles are rearranged as shown here:

71462.png

Use Thato's method to determine the number of tiles in the 23rd figure.


(c) Complete the flow diagram below by writing the appropriate operators so that it can be used to calculate the number of tiles in any figure of the pattern.

71453.png 

(d) How many tiles will there be in the 50th figure if the pattern is continued?


1. Write down the next four terms in each sequence. Also explain, in each case, how you figured out what the terms are.

(a) 2; 4; 8; 14; 22; 32; 44;

58; 74; 92; 112

Add consecutive even numbers: 2 then 4 then 6 and so on.

(b) 2; 6; 18; 54; 162;

486; 1 458; 4 374; 13 122

Multiply each number by 3 to find the next term

(c) 1; 7; 13; 19; 25;

58; 74; 92; 112

Add 6 to find the next term

2. (a) Complete the table below by calculating the missing terms.

Position in sequence

1

2

3

4

5

7

10

Term

3

10

17

(b) Write the rule to calculate the term from the position in the sequence in words.

Multiply the position number by 7 and subtract 4.

3. Consider the stacks below.

71385.png 

(a) How many cubes will there be in stack 5?

125

(b) Complete the table.

Stack number

1

2

3

4

5

6

10

Number of cubes

1

8

27

(c) Write down the rule to calculate the number of cubes for any stack number.

Cube the stack number, i.e. (stack number)3


In this chapter you will learn about quantities that change, such as the height of a tree. As the tree grows, the height changes. A quantity that changes is called a variable quantity or just a variable.

It is often the case that when one quantity changes, another quantity also changes. For example, as you make more and more calls on a phone, the total cost increases. In this case, we say there is a relationship between the amount of money you have to pay and the number of calls you make. You will learn how to describe a relationship between two quantities in different ways.

5.1 Constant and variable quantities 95

5.2 Different ways to describe relationships 100

5.3 Algebraic symbols for variables and relationships 103

Maths1_gr8_ch5_fig1.tif 

5 Functions and relationships

5.1 Constant and variable quantities

Looking for connections between quantities

Consider the following seven situations. There are two quantities in each situation. For each quantity, state whether it is constant (always the same number) or whether it changes. Also state, in each case, whether one quantity has an influence on the other. If it has, try to say how the one quantity will influence the other quantity.

1. Your age and the number of fingers on your hands




2. The number of calls you make and the airtime left on your cellphone




3. The length of your arm and your ability to finish Mathematics tests quickly




4. The number of identical houses to be built and the number of bricks required




5. The number of learners at a school and the length of the school day




6. The number of learners at a school and the number of classrooms needed



7. The number of matches in each arrangement here, and the number of triangles in the arrangement

51380.png 



If one variable quantity is influenced by another, we say there is a relationship between the two variables. It is sometimes possible to find out what value of the one quantity, in other words what number, is linked to a specific value of the other quantity.

A quantity that changes is called a variable quantity or just a variable.

8. (a) Look at the match arrangements in question 7. If you know that there are 3 triangles in an arrangement, can you say with certainty how many matches there are in that specific arrangement?



(b) How many matches are there in the arrangement with 10 triangles?


(c) Is there another possible answer for question (b)?


9. Complete the flow diagram by filling in all the missing numbers. Do you see any connections between the situation in question 7 and this flow diagram? If so, describe the connections.




51437.png

Completing some flow diagrams

A relationship between two quantities can be shown with a flow diagram, such as those below. Unfortunately, only some of the numbers can be shown on a flow diagram.

1. Calculate the output numbers for the flow diagram below. Some input numbers are missing. Choose and insert your own input numbers.

Each input number in a flow diagram has a corresponding output number. The first (top) input number corresponds to the first output number. The second input number corresponds to the second output number and so on.

We call +5 the operator.

(a)

51484.png 

(b) What type of numbers are the given input numbers?



(c) In the above flow diagram, the output number 8 corresponds to the input number 3. Complete the following sentences:

In the relationship shown in the above flow diagram, the output number


corresponds to the input number -1.

The input number


corresponds to the output number 7.

If more places are added to the flow diagram, the input number


will correspond to the output number 31.

2. (a) Complete this flow diagram.

51596.png 

(b) Compare this flow diagram to the flow diagram in question 1. What link do you find between the two?


3. Complete the flow diagrams below. You have to find out what the operator for (b) is, and fill it in yourself.

(a) (b)

51613.png 

(c) What number can you add in (a), instead of subtracting 5, that will produce the same output numbers?


(d) What number can you subtract in (b), instead of adding a number, that will produce the same output numbers?


4. Complete the flow diagram:

51633.png 

A completed flow diagram shows two kinds of information:

The flow diagram that you have completed in question 4 shows the following information:

Input numbers

-1

-2

-3

-4

-5

Output numbers

34

28

22

16

10

5. (a) Describe in words how the output numbers below can be calculated.

51714.png 






(b) Use the table below to show which output numbers are connected to which input numbers in the above flow diagram.

6. The following information is available about the floor space and cost of new houses in a new development. The cost of an empty stand is R180 000.

Floor space in square metres

90

120

150

180

210

Cost of house and stand

540 000

660 000

780 000

900 000

1 020 000

(a) Represent the above information on the flow diagram below.

60990.png 

(b) Show what the houses only will cost, if you get the stand for free.

60998.png 

(c) Try to figure out what the cost of a house and stand will be, if there are exactly one hundred 1 m by 1 m sections of floor space in the house.


5.2 Different ways to describe relationships

A relationship between red dots and blue dots

Here is an example of a relationship between two quantities:

51821.png 

In each arrangement there are some red dots and some blue dots.

1. How many blue dots are there if there is one red dot?


2. How many blue dots are there if there are two red dots?


3. How many blue dots are there if there are three red dots?


4. How many blue dots are there if there are four red dots?


5. How many blue dots are there if there are five red dots?


6. How many blue dots are there if there are six red dots?


7. How many blue dots are there if there are seven red dots?


8. How many blue dots are there if there are ten red dots?


9. How many blue dots are there if there are twenty red dots?


10. How many blue dots are there if there are one hundred red dots?


11. Which of the descriptions on the next page correctly describe the relationship between the number of blue dots and the number of red dots in the above arrangements? Test each description thoroughly for all the above arrangements. List them on the dotted line below. Write only the letters, for example (d).

Something to think about

Are there different possibilitiesfor the number of blue dots if there are 3 red dots in the arrangement?

Are there different possibili-ties for the number of blue dots if there are 2 red dots in the arrangement?

Are there different possibili-ties for the number of blue dots if there are 20 red dots in the arrangement?


(a) the number of red dots 51899.png the number of blue dots

(b) to calculate the number of blue dots you multiply the number of red dots by 2, add 1 and multiply the answer by 2

(c) number of blue dots = 2 \times the number of red dots + 4

(d)

Number of red dots

1

2

3

4

5

6

Number of blue dots

6

10

14

18

22

26

(e)

51937.png 

(f)

51946.png 

(g) number of blue dots = 4 \times the number of red dots + 2

(h) number of blue dots = 2 \times (2 \times the number of red dots + 1) (Remember that the calculations inside the brackets are done first.)

51910.png 

The descriptions in (c), (g) and (h) above are called word formulae.

Translating between different languages of description

Arelationship between two quantities can be described in different ways, including the following:

You will learn about symbolic formulae in section 5.3.

1. The relationship between two quantities is described as follows:

The second quantity is always 3 times the first quantity plus 8.

The first quantity varies between 1 and 5, and it is always a whole number.

(a) Describe this relationship with the flow diagram.

52112.png 

(b) Describe the relationship between the two quantities with this table.

(c) Describe the relationship between the two quantities with a word formula.



2. The relationship between two quantities is described as follows:

The input numbers are the first five odd numbers.

value of the one quantity

52166.png 

the corresponding value of the other quantity

(a) Describe this relationship with a table.

(b) Describe the relationship with a word formula.


5.3 Algebraic symbols for variables and relationships

Describing procedures in different ways

1. In each case do four things:

(a) input number 52187.png output number

Input number

5

10

15

20

25

30

Output number

the output number =





(b) input number 52222.png output number

Input number

5

10

15

20

25

30

Output number





(c) input number 52231.png output number

Input number

5

10

15

20

25

30

Output number




Formulae with symbols

Instead of writing "input number" and "output number" in formulae, one may just write a single letter symbol as an abbreviation.

Mathematicians have long ago adopted the convention of using the letter symbol x as an abbreviation for "input number", and the letter symbol y as an abbreviation for "output number".

Other letter symbols than x and y are also used to indicate variable quantities.

The word formula you wrote for question 1(a) can be written more shortly as

y = 10 \times x + 15

Mathematicians have also long ago agreed that one may leave the \times-sign out when writing symbolic formulae.

It is not at all wrong to use the multiplication sign in symbolic formulae.

So, instead of y = 10 \times x + 15 we may write y = 10x + 15.

2. Rewrite your word formulae in questions 1(b) and 1(c) as symbolic formulae.

y = 10(x + 15) y = 5(2x + 3)

3. Write a word formula for each of the following relationships:

(a) y = 7x + 10


(b) y = 7(x + 10)


(c) y = 7(2x + 10)


Writing symbolic FORMULAE

Describe each of the following relationships with a symbolic formula:

1. To calculate the output number, the input number is multiplied by 4 and 7 is subtracted from the answer.

y = 4x - 7

2. To calculate the output number, 7 is subtracted from the input number and the answer is multiplied by 5.



3. To calculate the output number, 7 is subtracted from the input number, the answer is multiplied by 5 and 3 is added to this answer.





An algebraic expression is a description of certain calculations that have to be done in a certain order. In this chapter, you will be introduced to the language of algebra. You will also learn about expressions that appear to be different but that produce the same results when evaluated. When we evaluate an expression, we choose or are given a value of the variable in the expression. Because now we have an actual value, we can carry out the operations (+, -, \times, \div) in the expression using this value.

6.1 Algebraic language 107

6.2 Add and subtract like terms 112

6 Algebraic expressions 1

6.1 Algebraic language

words, diagrams and symbols

1. Complete this table.

Words

Flow diagram

Expression

Multiply a number by two and add six to the answer.

57775.png 

2 \times x + 6

(a)

Add three to a number and then multiply the answer by two.

2 \times (x + 3)

(b)

Multiply a number by five and then subtract one from the answer.

57759.png 

5x - 1

(c)

Multiply a number by four and then add seven to the answer.

7 + 4 \times x

(d)

Multiply a number by negative five and then add ten to the answer.

10 - 5 \times x

An algebraic expression indicates a sequence of calculations that can also be described in words or with a flow diagram.

2. Write the following expressions in ‘normal' algebraic language:

(a) -2 \times a + b (b) a2

-2a + b

2a

Looking different but yet the same

1. Complete the table by calculating the numerical values of the expressions for the values of x. Some answers for x = 1 have been done for you as an example.

x

1

3

7

10

(a)

2x + 3x

2 \times 1 + 3 \times 1

2 + 3 = 5

2 \times 3 + 3 \times 3

6 + 9 = 15

2 \times 7 + 3 \times 7

14 + 21 = 35

2 \times 10 + 3 \times 10

20 + 30 = 50

(b)

5x

5 \times 1

5 \times 1 = 5

5 \times 3

5 \times 3 = 15

5 \times 7

5 \times 7 = 35

5 \times 10

5 \times 10 = 50

(c)

2x + 3

2 \times 1 + 3

2 + 3 = 5

2 \times 3 + 3

6 + 3 = 9

2 \times 7 + 3

14 + 3 = 17

2 \times 10 + 3

20 + 3 = 23

(d)

5x2

5 \times (1)2

5 \times 1 = 5

5 \times (3)2

5 \times 9 = 45

5 \times (7)2

5 \times 49 = 245

5 \times (10)2

5 \times 100 = 500

2. Do the expressions 2x + 3x and 5x, in question 1 above, produce different answers or the same answer for:

(a) x = 3? (b) x = 10?

same answer (15)

same answer (50)

3. Do the expressions 2x + 3 and 5x produce different answers or the same answer for:

(a) x = 3? (b) x = 10?

different answers (15 and 9)

different answers (50 and 23)

4. Write down all the algebraic expressions in question 1 that have the same numerical value for the same value(s) of x, although they may look different. Justify your answer.







One of the things we do in algebra is to evaluate expressions. When we evaluate an expression we choose or are given a value of the variable in the

expression. Because now we have an actual value we can carry out the operations in the expression using this value, as in the examples given in the table.

5. Say whether the following statements are true or false. Explain your answer in each case.

(a) The expressions 2x + 3x and 5x are equivalent.



(b) The expressions 2x + 3 and 5x are equivalent.




6. Consider the expressions 3x + 2z + y and 6xyz.

Remember that 6xyz is the same as 6 \times x \times y \timesz.

(a) What is the value of 3x + 2z + y for x = 4, y = 7 and z = 10?



(b) What is the value of 6xyz forx = 4, y = 7 and z = 10?



(c) Are the expressions 3x + 2z + y and 6xyz equivalent? Explain.


x, y and z.

To show that the two expressions in question 5(a) are equivalent we write 2x + 3x = 5x.

The term 3x is a product. The number 3 is called the coefficient of x.

We can explain why this is so:

2x + 3x = (x + x) + (x + x + x) = 5x

We say the expression 2x + 3x simplifies to 5x.

7. In each case below, write down an expression equivalent to the one given.

(a) 3x + 3x (b) 3x + 8x + 2x

6x

13x

(c) 8b + 2b + 2b (d) 7m + 2m + 10m

12b

19m

(e) 3x2 + 3x2 (f) 3x2 + 8x2 + 2x2

6x2

13x2

8. What is the coefficient of x2 for the expression equivalent to 3x2 + 8x2 + 2x2?



In an expression that can be written as a sum, the different parts of the expression are called the terms of the expression. For example, 3x, 2z and y are the terms of the expression 3x + 2z + y.

9. (a) Calculate the numerical value of 10x + 2y for x = 3 and y = 2 by completing the empty spaces in the diagram.

Input value: 3

57331.png

+


=


(Output value)

Input value: 2

(b) What is the output value for the expression 12xy for x = 3 and y = 2?


(c) Are the expressions 10x + 2y and 12xy equivalent? Explain.



(d) Are the terms 10x and 2y like or unlike terms? Explain.


10. (a) Which of the following algebraic expressions do you think will give the same results?

A. 6x + 4x B. 10x C. 10x2 D. 9x + x


(b) Test the algebraic expressions you have identified for the following values of x:

x = 10 x = 17 x = 54





































(c) Are the terms 6x and 4x like or unlike terms? Explain.



(d) Are the terms 10x and 10x2 like or unlike terms? Explain.





11. Ashraf and Hendrik have a disagreement about whether the terms 7x2y3 and 301y3x2 are like terms or not. Hendrik thinks they are not, because in the first term the x2 comes before the y3 whereas in the second term the y3 comes before the x2.

Explain to Hendrik why his argument is not correct.







12. Explain why the terms 5abc, 10acb and 15cba are like terms.



6.2 Add and subtract like terms

rearrange terms and then combine like terms

1. Complete the table by evaluating the expressions for the given values of x.

x

1

2

10

30x + 80

30 \times 1 + 80

= 30 + 80 = 110

30 \times 2 + 80

60 + 80 = 140

30 \times 10 + 80

300 + 80 = 380

5x + 20

5 \times 1 + 20

5 + 20 = 25

5 \times 2 + 20

10 + 20 = 30

5 \times 10 + 20

50 + 20 = 70

30x + 80 + 5x + 20

30 \times 1 + 80 +5 \times 1 + 20

= 30 + 80 + 5 + 20

= 135

30 \times 2 + 80 +5 \times 2 + 20

= 60 + 80 + 10 + 20

= 170

30 \times 10 + 80 +5 \times 10 + 20

= 300 + 80 + 50 + 20

= 450

35x + 100

35 \times 1 + 100

= 35 + 100 = 135

35 \times 2 + 100

= 70 + 100 = 170

35 \times 10 + 100

350 + 100 = 450

135x

135 \times 1

= 135

135 \times 2

= 270

135 \times 10

=1 350

2. Write down all the expressions in the table that are equivalent.


3. Tim thinks that the expressions 135x and 35x + 100 are equivalent because for x = 1 they both have the same numerical value 135.

Explain to Tim why the two expressions are not equivalent.




74558.png 

We have already come across the commutative and associative properties of operations. We will now use these properties to help us form equivalent algebraic expressions.

Commutative property

The order in which we add or multiply numbers does not change the answer: a + b = b + a and ab = ba

Associative property

The way in which we group three or more numbers when adding or multiplying does not change the answer: (a + b) + c = a + (b + c) and (ab)c = a(bc)

We can find an equivalent expression by rearranging and combining like terms, as shown below:

The terms 80 and 20 are called constants. The numbers 30 and 5 are called coefficients.

30x + 80 + 5x + 20

Hence 30x + (80 + 5x) + 20

Hence 30x + (5x + 80) + 20

= (30x + 5x) + (80 + 20)

= 35x + 100

Brackets are used in theexpression on the left to show how the like terms have been rearranged.

Like terms are combined to form a single term.

The terms 30x and 5x are combined to get the new term 35x, and the terms 80 and 20 are combined to form the new term 100. We say that the expression 30x + 80 + 5x + 20 is simplified to a new expression 35x + 100.

4. Simplify the following expressions:

(a) 13x + 7 + 6x - 2 (b) 21x - 8 + 7x + 15

= 13x + 6x + 7 - 2

= 21x + 7x - 8 + 15

= 19x + 5

= 28x + 7

(c) 18c - 12d + 5c - 7c (d) 3abc + 4 + 7abc - 6

= 18c + 5c - 7c - 12d

= 3abc + 7abc + 4 –6

= 16c - 12d

= 10abc - 2

(e) 12x2 + 2x - 2x2 + 8x (f) 7m3 + 7m2 + 9m3 + 1

= 12x2 - 2x2 + 2x + 8x

= 7m3 + 9m3 + 7m2 + 1

= 10x2 + 10x

= 16m3 + 7m2 + 1

When you are not sure about whether you correctly simplified an expression, it is always advisable to check your work by evaluating the original expression and the simplified expression for some values. This is a very useful habit to have.

5. Make a simpler expression that is equivalent to the given expression. Test your answer for three different values of x, and redo your work until you get it right.

(a) Simplify (15x + 7y) + (25x + 3 + 2y) (b) Simplify 12mn + 8mn

15x + 7y + 25x + 3 + 2y

12mn + 8mn

= 15x + 25x + 7y + 2y + 3

= 20mn

= 40x + 9y + 3

In questions 6 to 8 below, write down the letter representing the correct answer. Also explain why you think your answer is correct.

6. The sum of 5x2 + x + 7 and x - 9 is:

A. x2 - 2 B. 5x2 + 2x +16 C. 5x2 + 16 D. 5x2 + 2x - 2


7. The sum of 6x2 - x + 4 and x2 - 5 is equivalent to:

A. 7x2 - x + 9 B. 7x2 - x - 1 C. 6x4 - x - 9 D. 7x4 - x - 1


8. The sum of 5x2 + 2x + 4 and 3x2 - 5x - 1 can be expressed as:

A. 8x2 + 3x + 3 B. 8x2 + 3x - 3 C. 8x2 - 3x + 3 D. 8x2 - 3x - 3


56274.png 

Combining like terms is a useful algebraic habit. It allows us to replace an expression with another expression that may be convenient to work with.

Do the following questions to get a sense of what we are talking about.

convenient replacements

1. Consider the expression x + x + x + x + x + x + x + x + x + x. What is the value of the expression in each of the following cases?

(a) x = 2 (b) x = 50

x + x + x + x + x + x + x + x + x + x = 10x


2. Consider the expression x + x + x + z + z + y. What is the value of the expression in each of the following cases?

(a)x = 4, y = 7, z = 10 (b) x = 0, y = 8, z = 22

x + x + x + z + z + y = 3x + 2z + y




3. Suppose you have to evaluate 3x + 7x for x = 20. Will calculating 10 \times 20 give the correct answer? Explain.







Suppose we evaluate the expression 3x + 7x for x = 20 without first combining the like terms. We will have to do three calculations, namely 3 \times 20, then 7 \times 20 and then find the sum of the two: 3 \times 20 + 7 \times 20 = 60 + 140 = 200.

But if we first combine the like terms 3x and 7x into one term 10x, we only have to do one calculation: 10 \times 20 = 200. This is one way of thinking about the convenience or usefulness of simplifying an algebraic expression.

4. The expression 5x + 3x is given and you are required to evaluate it for x = 8. Will calculating 8 \times 8 give the correct answer? Explain.







5. Suppose you have to evaluate 7x + 5 for x = 10. Will calculating 12 \times 10 give the correct answer? Explain.





6. The expression 5x + 3 is given and you have to evaluate it for x = 8. Will calculating 8 \times 8 give the correct answer? Explain.



Samantha was asked to evaluate the expression 12x2 + 2x - 2x2 + 8x for x = 12. She thought to herself that just substituting the value of x directly into the terms would require a lot of work. She first combined the like terms as shown below:

The terms +2x and –2x2 change positions by the commutative property of operations.

12x2 - 2 56248.jpg 2 + 2 56246.jpg +8x

= 10x2 + 10x

Then for x = 10, Samantha found the value of 10x2 + 10x by calculating

10 \times 102 + 10 \times 10

= 1 000 + 100

= 1 100

Use Samatha's way of thinking for questions 7 to 9.

7. What is the value of 12x + 25x + 75x + 8x when x = 6?




8. Evaluate 3x2 + 7 + 2x2 + 3 for x = 5.




9. When Zama was asked to evaluate the expression 2n - 1 + 6n for n = 4, she wrote down the following:

2n - 1 + 6n = n + 6n = 6n2

Hence for n = 4: 6 \times (4)2 = 6 \times 8 = 48

Explain where Zama went wrong and why.







1. Complete the table.

Words

Flow diagram

Expression

(a)

Multiply a number by three and add two to the answer.

56235.png 

3x + 2

(b)

Multiply a number by nine and subtract six from the answer.

9x - 6

(c)

Multiply a number by seven and subtract three from the answer.

56215.png 

7x – 3

2. Which of the following pairs consist of like terms? Explain.

A. 3y; -7y B. 14e2; 5e C. 3y2z; 17y2z D. -bcd; 5bd

A and C. In A, the variable is y for both terms and is raised to the same power.

In C, the variables are y and z; y is raised to the power 2 and z to the power 1.

3. Write the following in the ‘normal' algebraic way:

(a) c2 + d3 (b) 7 \times d \times e \times f

2c + 3d

7def

4. Consider the expression 12x2 - 5x + 3.

(a) What is the number 12 called?

the coefficient of x2

(b) Write down the coefficient of x.

-5

(c) What name is given to the number 3?

a constant

5. Explain why the terms 5pqr and -10prq and 15qrp are like terms.

All three terms have exactly the same variables, raised to the same power.

6. If y = 7, what is the value of each of the following?

(a) y + 8 (b) 9y (c) 7 - y

7 + 8 = 15

9 \times 7 = 63

7 - 7 = 0

7. Simplify the following expressions:

(a) 18c + 12d + 5c - 7c (b) 3def + 4 + 7def - 6

= 18c + 5c - 7c - 12d

= 3def + 7def + 4 - 6

= 16c - 12d

= 10def - 2

8. Evaluate the following expressions for y = 3, z = -1:

(a) 2y2 + 3z (b) (2y)2 + 3z

2 \times (3)2 + 3 \times (-1)

(2 \times 3)2 + 3(-1)

= 2 \times 9 + (-3)

= (6)2 - 3

= 18 - 3 = 15

= 36 - 3 = 33

9. Write each algebraic expression in the simplest form.

(a) 5y + 15y (b) 5c + 6c - 3c + 2c

20y

10c

(c) 4b + 3 + 16b - 5 (d) 7m + 3n + 2 - 6m

20b- 2

m + 3n + 2

(e) 5h2 + 17 - 2h2 + 3 (f) 7e2f + 3ef + 2 + 4ef

3h2 + 20

7e2f + 7ef + 2

10. Evaluate the following expressions:

(a) 3y+ 3y + 3y + 3y + 3y + 3y for y = 18

3y+ 3y + 3y + 3y + 3y + 3y = 18y

18 \times 18 = 324 for y = 18

(b) 13y + 14 - 3y + 6 for y = 200

13y + 14 - 3y + 6 = 10y + 20

10 \times 200 + 20 = 2 000 + 20 = 2 020 for y = 20

(c) 20 - y2 + 101y2 + 80 for y = 1

20 - y2 + 101y2 + 80 = 100y2 + 100

100 \times (1)2 + 100 = 100 + 100 = 200 for y = 1

(d) 12y2 + 3yz + 18y2 + 2yz for y = 3 and z = 2

12y2 + 3yz + 18y2 + 2yz = 30y2 + 5yz

30 \times (3)2 + 5 \times 3 \times 2 = 30 \times 9 + 30 = 270 + 30 = 300

gr8ch6.tif

In this chapter you will learn to find numbers that make certain statements true. A statement about an unknown number is called an equation. When we work to find out which number will make the equation true, we say we solve the equation. The number that makes the equation true is called the solution of the equation.

7.1 Setting up equations 121

7.2 Solving equations by inspection 123

7.3 More examples 124

7 Algebraic equations 1

7.1 Setting up equations

An equation is a mathematical sentence that is true for some numbers but false for other numbers. The following are examples of equations:

x + 3 = 11 and 2x = 8

x + 3 = 11 is true if x = 8, but false if x = 3.

looking for numbers to make statements true

1. Are the following statements true or false? Justify your answer.

(a) x - 3 = 0, if x = -3


(b) x3 = 8, if x = -2


(c) 3x = -6, if x = -3


(d) 3x = 1, if x = 1


(e) 6x + 5 = 47, if x = 7



2. Find the original number. Show your reasoning.

(a) A number multiplied by 10 is 80.


(b) Add 83 to a number and the answer is 100.


(c) Divide a number by 5 and the answer is 4.


(d) Multiply a number by 4 and the answer is 20.


(e) Twice a number is 100.


(f) A number raised to the power 5 is 32.


(g) A number raised to the power 4 is -81.


(h) Fifteen times a number is 90.


(i) 93 added to a number is -3.


(j) Half a number is 15.


3. Write the equations below in words using "a number" in place of the letter symbol x. Then write what you think "the number" is in each case.

Example: 4 + x = 23. Four plus a number equals twenty-three. The number is 19.

(a) 8x = 72


(b) 60488.png= 2


(c) 2x + 5 = 21


(d) 12 + 9x = 30


(e) 30 - 2x = 40


(f) 5x + 4 = 3x + 10



7.2 Solving equations by inspection

THE ANSWER IS IN PLAIN SIGHT

1. Seven equations are given below the table. Use the table to find out for which of the given values of x it will be true that the left-hand side of the equation is equal to the right-hand side.

You can read the solutions of an equation from a table.

x

-3

-2

-1

0

1

2

3

4

2x + 3

-3

-1

1

3

5

7

9

11

x + 4

1

2

3

4

5

6

7

8

9 - x

12

11

10

9

8

7

6

5

3x - 2

-11

-8

-5

-2

1

4

7

10

10x - 7

-37

-27

-17

-7

3

13

23

33

5x + 3

-12

-7

-2

3

8

13

18

23

10 - 3x

19

16

13

10

7

4

1

-2

(a) 2x + 3 = 5x + 3 (b) 5x + 3 = 9 - x

x = 0

x = 1

(c) 2x + 3 = x + 4 (d) 10x - 7 = 5x + 3

x = 1

x = 2

(e) 3x - 2 = x + 4 (f) 9 - x = 2x + 3

x = 3

x = 2

(g) 10 - 3x = 3x - 2

x = 2

2. Which of the equations in question 1 have the same solutions? Explain.




3. Complete the table below. Then answer the questions that follow.

You can also do a search by narrowing down the possible solution to an equation.

x

0

5

10

15

20

25

30

35

40

2x + 3

3

13

23

33

43

53

63

73

83

3x - 10

-10

5

20

35

50

65

80

95

110

(a) Can you find a solution for 2x + 3 = 3x - 10 in the table?


(b) What happens to the values of 2x + 3 and 3x - 10 as x increases? Do they become bigger or smaller?


(c) Is there a point where the value of 3x - 10 becomes bigger or smaller than the value of 2x + 3 as the value of x increases? If so, between which x-values does this happen?

This point where the two expressions are equal is called the break-even point.


(d) Now that you narrowed down where the possible solution can be, try other possible values for x until you find out for what value of x the statement 2x + 3 = 3x - 10 is true.

"Searching" for the solution of an equation by using tables or by narrowing down to the possible solution is called solution by inspection.




74451.png 

7.3 More examples

looking for and checking solutions

1. What is the solution for the equations below?

(a) x - 3 = 4 (b) x + 2 = 9

x = 7

x = 7

(c) 3x = 21 (d) 3x + 1 = 22

x = 7

x = 7

When a certain number is the solution of an equation we say that the number satisfies the equation. For example, x = 4 satisfies the equation 3x = 12 because 3 \times 4 = 12.

2. Choose the number in brackets that satisfies the equation. Explain your choice.

(a) 12x = 84 {5; 7; 10; 12}


(b) 58867.png = 12 {-7; 0; 7; 10}


(c) 48 = 8k + 8 {-5; 0; 5; 10}


(d) 19 - 8m = 3 {-2; -1; 0; 1; 2}


(e) 20 = 6y - 4 {3; 4; 5; 6}


(f) x3 = -64 {-8; -4; 4; 8}


(g) 5x =125 {-3; -1; 1; 3}


(h) 2x = 8 {1; 2; 3; 4}


(i) x2 = 9 {1; 2; 3; 4}



3. What makes the following equations true? Check your answers.

(a) m + 8 = 100 (b) 80 = x + 60

m = 92

x = 20

92 + 8 = 100

20 + 60 = 80

(c) 26 - k = 0 (d) 105 \times y = 0

k = 26

y = 0

26 - 26 = 0

105 \times 0 = 0

(e) k \times10 = 10 (f) 5x = 100

k = 1

x = 20

1 \times 10 = 10

5 \times 20 = 100

(g) 58559.png = 5 (h) 3 = 58552.png

t = 3

t = 15

= 5

= 3

4. Solve the equations below by inspection. Check your answers.

(a) 12x + 14 = 50 (b) 100 = 15m + 25

12x = 36

75 = 15m

x = 3

m = 5

12 \times 3 + 14 = 36 + 14 = 50

15 \times 5 + 25 = 75 + 25 = 100

(c) 58302.png = 20 (d) 7m + 5 = 40

100 = 20x

7m = 35

x = 5

m = 5

= 100 \div 5 = 20

7 \times 5 + 5 = 35 + 5 = 40

(e) 2x + 8 = 10 (f) 3x + 10 = 31

2x = 2

3x = 21

x = 1

x = 7

(g) -1 + 2x = -11 (h) 2 + 73812.png = 5

2x = -10

= 3

x = -5

x = 21

(i) 100 = 64 + 9x (j) 73707.png = 4

36 = 9x

x = 4 \times

x = 4

x = 12


gr8ch7.tif

Revision 128

Assessment 140

Revision

Show all your steps of working.

78975.png 

whole numbers

1. (a) Write both 300 and 160 as products of prime factors.



(b) Determine the HCF and LCM of 300 and 160.




2. Tommy, Thami and Timmy are given birthday money by their grandmother in the ratio of their ages. They are turning 11, 13 and 16 years old, respectively. If the total amount of money given to all three boys is R1 000, how much money does Thami get on his birthday?




3. Tshepo and his family are driving to the coast on holiday. The distance is 1 200 km and they must reach their destination in 12 hours. After 5 hours, they have travelled 575 km. Then one of their tyres bursts. It takes 45 minutes to get the spare wheel on, before they can drive again. At what average speed must they drive the remainder of the journey to reach their destination on time?





4. The number of teachers at a school has increased in the ratio 5 : 6. If there used to be 25 teachers at the school, how many teachers are there now?




5. ABC for Life needs to have their annual statements audited. They are quoted R8 500 + 14% VAT by Audits Inc. How much will ABC for Life have to pay Audits Inc. in total?






6. Reshmi invests R35 000 for three years at an interest rate of 8,2% per annum. Determine how much money will be in her account at the end of the investment period.







7. Lesebo wants to buy a lounge suite that costs R7 999 cash. He does not have enough money and so decides to buy it on hire purchase. The store requires a 15% deposit up front, and 18 monthly instalments of R445.

(a) Calculate the deposit that Lesebo must pay.





(b) How much extra does Lesebo pay because he buys the lounge suite on hire purchase, rather than in cash?






8. Consider the following exchange rates table:

South African Rand

1.00 ZAR

inv. 1.00 ZAR

Euro

0.075370

13.267807

US Dollar

0.098243

10.178807

British Pound

0.064602

15.479409

Indian Rupee

5.558584

0.179902

Australian Dollar

0.102281

9.776984

Canadian Dollar

0.101583

9.844200

Emirati Dirham

0.360838

2.771327

Swiss Franc

0.093651

10.677960

Chinese Yuan Renminbi

0.603065

1.658195

Malaysian Ringgit

0.303523

3.294646

(a) Write down the amount in rand that needs to be exchanged to get 1 Swiss franc. Give your answer to the nearest cent.


(b) Write down the only currency for which an exchange of R100 will give you more than 100 units of that currency.



(c) Ntsako is travelling to Dubai and converts R10 000 into Emirati dirhams. How many dirhams does Ntsako receive (assume no commission)?



integers

Don't use a calculator for any of the questions in this section.

1. Write a number in each box to make the calculations correct.

(a)

+

= -11 (b)

-

= -11

2. Fill <, > or = into each block to show the relationships.

(a) -23

20 (b) -345

-350

(c) 4 - 3

3 - 4 (d) 5 - 7

-(7 - 5)

(e) -9 \times 2

-9 \times -2 (f) -4 \times 5

4 \times -5

(g) -10 \div 5

-10 \div -2 (h) -15 \times -15

224

3. Follow the pattern to complete the number sequences.

(a) 8; 5; 2;

(b) 2; -4; 8;

(c) -289; -293; -297;

4. Look at the number lines. In each case, the missing number is halfway between the other two numbers. Fill in the correct values in the boxes.

(a) (b)

78588.png
78595.png

5. Calculate the following:

(a) -5 - 7 (b) 7 - 10

-12

-3

(c) 8 - (-9) (d) (-5)(-2)(-4)

17

-40

(e) 5 + 4 \times -2 (f) 78479.png 

-3

(2 \times 4) \div -4 = 8 \div -4 = -2

(g) 78420.png (h) 78412.png 

= 79198.jpg

= 79206.jpg

= -5

= 1

6. (a) Write down two numbers that multiply to give -15. (One of the numbers must be positive and the other negative.)


(b) Write down two numbers that add to 15. One of the numbers must be positive and the other negative.


7. At 5 a.m., the temperature in Kimberley was -3 °C. At 1 p.m., it was 17 °C. By how many degrees had the temperature risen?



8. A submarine is 220 m below the surface of the sea. It travels 75 m upwards. How far below the surface is it now?




exponents

You should not use a calculator for any of the questions in this section.

1. Write down the value of the following:

(a) (-3)3


(b) -52


(c) (-1)200


(d) (102)2


2. Write the following numbers in scientific notation:

(a) 200 000 (b) 12,345

2 \times 105

1,2345 \times 101

3. Write the following numbers in ordinary notation:

(a) 1,3 \times 102 (b) 7,01 \times 107

130

70 100 000

4. Which of the following numbers is bigger: 5,23 \times 1010 or 2,9 \times 1011?





5. Simplify the following:

(a) 27 \times 23


(b) 2x3 \times 4x4


(c) (-8y6) \div (4y3)


(d) (3x8)3


(e) (2x5)(0,5x-5)


(f) (-3a2b3c)(-4abc2)2


(g) 77993.png


6. Write down the values of each of the following:

(a) (0,6)2


(b) (0,2)3


(c) 77913.png


(d) 77879.png


(e) 4 77848.png


(f) 77817.png


numeric and geometric patterns

1. For each of the following sequences, write the rule for the relationship between each term and the following term in words. Then use the rule to write the next three terms in the sequence.

(a) 12; 7; 2;


;


;



(b) -2; -6; -18;


;


;



(c) 100; -50; 25;


;


;



(d) 3; 4; 7; 11;


;


;




2. In this question, you are given the rule by which each term of the sequence can be found. In all cases, n is the position of the term.

Determine the first three terms of each of the sequences. (Hint: Substitute n = 1 to find the value of the first term.)

(a) n\times 4


(b) n\times 5 - 12


(c) 2 \times n2


(d) 3n\div 3 \times -2


3. Write down the rule by which each term of the sequence can be found (in a similar format to those given in question 2, where n is the position of the term).

(a) 2; 4; 6; …


(b) –7; –3; 1; ...


(c) 2; 4; 8; …


(d) 9; 16; 23; …


4. Use the rules you have found in question 3 to find the value of the 20th term of the sequences in questions 3(a) and 3(b).

(a)



(b)




5. Find the relationship between the position of the term in the sequence and the value of the term, and use it to fill in the missing values in the tables.

(a)

Position in sequence

1

2

3

4

25

Value of the term

-8

-11

-14

-17

-80

(b)

Position in sequence

1

2

3

6

10

Value of the term

1

3

9

243

19 683

6. The image below shows a series of patterns created by matches.

77426.png 

(a) Write in words the rule that describes the number of matches needed for each new pattern.



(b) Use the rule to determine the missing values in the table below, and fill them in.

Number of the pattern

1

2

3

4

20

50

Number of matches needed

4

7

10

13

61

151




functions and relationships

1. Fill in the missing input values, output values or rule in these flow diagrams. Note that p and t are integers.

(a)

77406.png 

(b)

77397.png 

(c)

95229.png

2. Consider the values in the following table. The rule for finding y is: divide x by -2 and subtract 4. Use the rule to determine the missing values in the table, and write them in.

x

-6

-2

0

2

5

-104

y

-1

-3

-4

-5

-6,5

48



3. Consider the values in the following table:

x

-2

-1

0

1

2

4

15

28

y

1

3

5

7

9

13

35

61

(a) Write in words the rule for finding the y-values in the table.



(b) Use the rule to determine the missing values in the table, and write them in.


algebraic expressions 1

1. Look at this algebraic expression: 5x3 - 9 + 4x - 3x2.

(a) How many terms does this expression have?


(b) What is the variable in this expression?


(c) What is the coefficient of the x2 term?


(d) What is the constant in this expression?


(e) Rewrite the expression so that the terms are in order of decreasing powers of x.


2. In this question, x = 6 and y = 17. Complete the rules to show different ways to determine y if x is known. The first way is done for you:

Way 1: Multiply x by 2 and add 5. This can be written as y = 2x + 5

(a) Way 2: Multiply x by


and then subtract


. This can be written as

y = 2x - (-5)

(b) Way 3: Divide x by


and then add


. This can be written as

y = x \div 0,5 + 5

(c) Way 4: Add


to x, and then multiply by


. This can be written as

y = 2(x + 2,5)

3. Simplify:

(a) 2x2 + 3x2


(b) 9xy - 12yx


(c) 3y2 - 4y + 3y - 2y2


(d) 9m3 + 9m2 + 9m3 - 3


4. Calculate the value of the following expressions if a = -2; b = 3; c = -1 and d = 0:

(a) abc


(b) -a2


(c) (abc)d


(d) a + b - 2c


(e) (a + b)10


algebraic equations 1

1. Write equations that represent the given information:

(a) Nandi is x years old. Shaba, who is y years old, is three years older than Nandi.

y = x + 3

(b) The temperature at Colesberg during the day was x °C. But at night, the temperature dropped by 15 degrees to reach -2 °C.


2. Solve the following equations for x:

(a) x + 5 = 2 (b) 7 - x = 9

x + 5 - 5 = 2 - 5

7 - x - 7 = 9 - 7

x = -3

x = -2

(c) 3x - 1 = -10 (d) 2x3 = -16



x3 = -8




x = -2


(e) 2x = 16 (f) 2(3)x = 6





x = 4




3. If 3n - 1 = 11, what is the value of 4n?




4. If c = a + b and a + b + c = 16, determine the value of c.


c + c = 16


5. (a) If 2a + 3 = b, write down values for a and b that will make the equation true.


(b) Write down a different pair of values to make the equation true.



Assessment

In this section, the numbers in brackets at the end of a question indicate the number of marks the question is worth. Use this information to help you determine how much working is needed. The total number of marks allocated to the assessment is 60.

76687.png 

1. The profits of GetRich Inc. have decreased in the ratio 5 : 3 due to the recession in the country. If their profits used to be R1 250 000, how much are their profits now? (2)




2. Which car has the better rate of petrol consumption: Ashley's car, which drove 520 km on 32 â„“ of petrol, or Zaza's car, which drove 880 km on 55 â„“ of petrol? Show all your working. (3)





3. Hanyani took out a R25 000 loan from a lender that charges him 22% interest each year. How much will he owe in one year's time? (3)




4. Consider the following exchange rates table:

South African Rand

1.00 ZAR

inv. 1.00 ZAR

Indian Rupee

5.558584

0.179902

Australian Dollar

0.102281

9.776984

Canadian Dollar

0.101583

9.844200

Emirati Dirham

0.360838

2.771327

Chinese Yuan Renminbi

0.603065

1.658195

Malaysian Ringgit

0.303523

3.294646

Chen returns from a business trip to Malaysia with 2 500 ringgit in his wallet. If he changes this money into rand in South Africa, how much will he receive? (2)



5. Fill <, > or = into the block to show the relationship between the number expressions:

(a) 6 - 4

4 - 6 (1)

(b) 2 \times -3

-32 (1)

6. Look at the number sequence below. Fill in the next term into the block.

-5; 10; -20;

(1)

7. Calculate the following:

(a) (-4)2 - 20 (2)


(b) 76594.png + 14 \div 2 (2)


8. Julius Caesar was a Roman emperor who lived from 100 BC to 44 BC. How old was he when he died? (2)


9. (a) Write down two numbers that divide to give an answer of -8. One of the numbers must be positive, and the other negative. (1)


(b) Write down two numbers that subtract to give an answer of 8. One of the numbers must be positive and the other negative. (1)


10. Write the following number in scientific notation: 17 million. (2)



11. Which of the following numbers is bigger: 3,47 \times 1021 or 7,99 \times 1020? (1)




12. Simplify the following, leaving all answers with positive exponents:

(a) 37 \times 3–2


(1)

(b) (-12y8) \div (-3y2)


(2)

(c) 76538.png


(4)

13. Write down the values of each of the following:

(a) (0,3)3


(1)

(b) 8 76483.png


(2)

14. Consider the following number sequence: 2; -8; 32; …

(a) Write in words the rule by which each term of the sequence can be found. (1)


(b) Write the next three terms in this sequence. (2)


15. The picture below shows a series of patterns created by matches.

76449.png 

(a) Write a formula for the rule that describes the relationship between the number of matches and the position of the term in the sequence (pattern number). Let n be the position of the term. (2)



(b) Use the rule to determine the values of a to c in the following table: (3)

Number of the pattern

1

2

3

4

15

c

Number of matches needed

8

15

22

a

b

148

a = 7n + 1 = 7 \times 4 + 1 = 29

b = 7n + 1 = 7 \times 15 + 1 = 106


16. Consider the values in the following table:

x

-2

-1

0

1

2

5

12

33

y

-7

-4

-1

2

5

14

35

98

(a) Write in words the rule for finding the y-values in the table. (2)



(b) Use the rule to determine the missing values in the table, and fill them in. (3)




17. Simplify:

(a) 2z2 - 3z2


(1)

(b) 8y2 - 6y + 4y - 7y2


(2)

18. Determine the value of 2a2 - 10 if a = -2. (2)




19. If c + 2d = 27, give the value of the following:

(a) 2c + 4d


(1)

(b) 76346.png


(1)

(c) 76314.png


(1)

20. Solve the following forx: (5)

(a) -x = -11 (b) 2x - 5 = -11 (c) 4x3 = 32

x = 11

2x = -6

x3 = 8

x = -3

x3 = 23